GIFT OF 
 
 ASTRONOMY 
 LIBRARY 
 
TABLES OF MINOR PLANETS DISCOVERED BY 
 
 JAMES C. WATSON 
 
 PART II 
 
 ON v. ZEIPEL'S THEORY OF THE PERTURBATIONS 
 OF THE MINOR PLANETS OF THE HECUBA GROUP 
 
MEMOIRS 
 
 or THK 
 
 NATIONAL ACADEMY OF SCIENCES 
 
 THIRD MEMOIR 
 
 WASHINGTON 
 
 GOVEBNMENT PRINTING OFFICE 
 1922 
 
 \ 
 
BHIOMJSM 
 
 SIM 
 
 10'YMadAOA JAMOITAM 
 
 V1X 
 
 cranrr 
 
 // 
 
 0DIIT<:- 
 
06377 
 
 ASTRONOMY 
 LIBRARY 
 
 NATIONAL ACADEMY OF SCIENCES. 
 
 Volume XIV. 
 
 THIRD MEMOIR. 
 
 TABLES OF MINOR PLANETS DISCOVERED BY 
 
 JAMES C. WATSON. 
 
 PARTIL 
 
 ON v. ZEIPEL'S THEORY OP THE PERTURBATIONS OP THE 
 MINOR PLANETS OF THE HECUBA GROUP. 
 
 BY 
 
 ARMIN O. LEUSCHNER, ANNA ESTELLE CLANCY, AKD 
 " SOPHIA H. LEVY. 
 
 50f>877 
 
.VI /C 'Hi i ii foV 
 
 UK >M:rt !/. (I il J HT 
 
 r io 
 
 .TI Til//! 
 
 airr io aworrAa.H'rwia'i airr r io YHOSHT pAiasis-s .-/ 
 
 airr M) 
 
 .YOW/uI, r J HJJHTgft /.W/A. ..JIM/ID^Tr-fJ -O 7-It 
 .Y/C'KI .H /.ll!'|i() 
 
CONTENTS. 
 
 Page. 
 
 Preface 7 
 
 Introduction g 
 
 I. Formulae and tables for the Hecuba group, according to the theory of Bohlin-v. Zeipel. and an example of their 
 
 se 10 
 
 Determination of constant elements and of perturbations of the mean anomaly 10 
 
 Perturbations of the radius vector 20 
 
 Perturbations of the third coordinate 21 
 
 Check computation 22 
 
 Computation of the perturbations for the time t 22 
 
 Comparison of the revised with v. Zeipel's original tables 27 
 
 Table A 28 
 
 Table B 30 
 
 Table C 31 
 
 Table D 34 
 
 Table E, 35 
 
 Table E 2 35 
 
 Table F 36 
 
 Table G 38 
 
 II. Tables for the determination of the perturbations of the Hecuba group of minor planets 41 
 
 Development of the differential equations for Wand for the third coordinate 41 
 
 Integration of the differential equation for W. 78 
 
 Comparison of tables 120 
 
 Perturbations of the mean anomaly 121 
 
 Comparison of tables 134 
 
 Perturbations of the radius vector 137 
 
 Perturbations of the third coordinate 140 
 
 Comparison of tables 146 
 
 Constants of integration in nSz and v 146 
 
 Comparison of tables 155 
 
 Erata in " Angenaherte Jupiter-Storungen fur die flecufco-Gruppe," H. v. Zeipel 156 
 
 Erata in ' ' Sur le Developpement des Perturbations Planetaires, " 1-7 and Tables I-XX, Karl Bohlin 157 
 
 5 
 
!*T II 
 
V -all \0 i'uutiq vflts >.)} 
 
 PREFACE. 
 
 Part I of "Tables of Minor Planets Discovered by James C. Watson," containing tables 
 for 12 of the 22 Watson planets, was published in 1910 in the Memoirs of the National Academy 
 of Sciences, Volume X, Seventh Memoir, with a preface by Simon Newcomb, in which he 
 gives an account of the early history of the investigations of the perturbations of the Watson 
 planets under the auspices of the Board of Trustees of the Watson Fund. 
 
 In the introduction to Part 1 1 reference is made to the Watson planets of the Hecuba 
 group, for which it was found necessary to construct special tables on the plan of Bohlin's 
 tables for the group 1/3. A comparison of these tables with similar tables by v. Zeipel remained 
 to be made before applying either of them to the development of perturbations of planets 
 of the Hecuba group. This comparison was completed in 1913 with the assistance of Miss A. 
 Estelle Glancy and Miss Sophia H. Levy, with the results set forth in the following pages. 
 
 Publication of these results was delayed, partly because it seemed desirable to verify the 
 tables by application to a number of planets and partly on account of interruptions caused in 
 recent years by war conditions. Miss Glancy, in particular, had undertaken to test the accuracy 
 of our tables, which we had applied to v. Zeipel's example, (10) Hygiea, by further investi- 
 gations on this example after joining the Observatorio Nacional at C6rdoba in 1913. This 
 test has now been completed with highly satisfactory results. The tables have also been 
 successfully applied to the Watson planets of the Hecuba group, including (175) Andromache, 
 which, on account of unusually large perturbations and other unfavorable conditions, forms 
 so far the most striking example of the applicability of the Bohlin-v. Zeipel method and of our 
 revised tables for the Hecuba group. 
 
 The plan of work included conferences, in which Miss Glancy and Miss Levy took a leading 
 part, for the discussion of the Bohlin-v. Zeipel method, involving verification of all mathe- 
 matical developments and formulation of plans for the construction of tables, and, after the 
 appearance of v. Zeipel's tables, for the comparison of v. Zeipel's original, and OUT revised tables. 
 The numerical work was carried out by Miss Glancy and Miss Levy, who have also contributed 
 very largely to the theoretical part of the work, and have prepared the principal details of the 
 manuscript. 
 
 To avoid confusion v. Zeipel's notation and method of procedure have been followed 
 throughout in completing our tables for the Hecuba group, which were well under way when 
 v. Zeipel's memoir appeared. 
 
 To aid computers in the use of the formulae and of the revised tables, Miss Glancy has 
 prepared detailed directions illustrated by an application to (10) Hygiea, the example first 
 chosen by v. Zeipel. These are contained in the first section of the present memoir. 
 
 Miss Glancy's contributions to this investigation and her work on (10) Hygiea were accepted 
 by the University of California in partial fulfillment of the requirements for the degree of 
 doctor of philosophy. 
 
 Miss Levy's contributions and her work on (175) Andromache were similarly accepted for 
 the same degree. 
 
 It seems highly desirable to make the tables for the development of the perturbations of 
 minor planets of the Hecuba group at once available to astronomers. They are therefore 
 published herewith, in advance of the perturbations and tables of the remaining Watson planets, 
 as Part II of "Tables of Minor Planets Discovered by James C. Watson." One or two parts, 
 which are to follow, will contain all the numerical results for the perturbations and tables of 
 Watson planets not published in Part I (1910). 
 
 This memoir is presented in two sections. The first section, entitled "Formulae and Tables 
 for the Hecuba Group, according to the Theory of Bohhn-v. Zeipel, and an Example of their 
 
 Pp. 200-201. 
 
8 
 
 PREFACE. 
 
 [Voi. xiv. 
 
 Use," contains a collection of the formulae to be used for any planet of the Hecuba group, the 
 general tables of the perturbations which must be employed, and a more complete application 
 of the formulae and the revised tables to the plane* (10) Hygiea, than v. Zeipel gives. The 
 second and more extensive section, entitled "Tables for the Determination of the Perturbations 
 of the Hecuba Group of Minor Planets," concerns the construction of the tables and their dis- 
 cussion with reference to the corresponding tables by v. Zeipel. It forms the preliminary part 
 of the in restigation but is presented last as supplementary to the final results given in the 
 first section. 
 
 In the second section the tabular values which differ from the corresponding numbers in 
 v. Zeipel's tables are placed in brackets. The general Tables XXXV, XXXVIII, XLIII, 
 LIV, LVi, LVn, LVI, LVII, of the second section, which, in order, are required to compute 
 the perturbations of any planet of the Hecuba group, are repeated without brackets at the 
 end of the first section as Tables A, B, C, D, E 1; E 2 , F, G, so that the first section is complete 
 in itself for use in developing the perturbations of any planet of this group without the necessity 
 of reference to the second section. 
 
 A general account of the investigations of the perturbations of the Watson planets was 
 presented to the Academy on April 16, 1916, and is published in the "Proceedings of the 
 National Academy of Sciences," Volume 4, No. 12, March, 1919. 
 
 ' ARMIN O LEUSCHNER. 
 
 -;.' WASHINGTON, D. C., 1918, December. 
 
 fun 
 
 -v?*lmH II-.r# 
 
 ' > 10(1 ill q i i!.'-- 1 u f 9nj <>! j.-iiiijur. '.'1 
 ^luAiovjiluu T.iio brut -(Ha i i:'iifjT)q o^i isi '{iliirifeinii' 
 !i wqi>\ .v-nilfloil -.tiij ")o vjilitia^Jqqu 9iij To ^> 
 
 v/7'i.l ?*\V. brui 
 
 /ni 
 
 /fmi Avw 1o nul 'i.'!T 
 
 e'jB<j moo oij io 
 
 li) *-.iM vd )uo Lom/i j !-/. >iio* 
 
 ini him iioitaJ 
 
 wiwn'tt -uiJ -ml 
 
 ^; hoc, ; 
 
 .v 
 
 nuc. ^n 
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 -ulj m t 
 
 i>io% r 
 y iii Jiii.> 
 
 lo anoiJadiiilT/q 
 
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TABLES OF MINOR PLANETS DISCOVERED BY JAMES C. WATSON. 
 
 By AHUM O. LKUSOTNZR, ANNA ESTELLB GLANCT, AND SOPHIA H. LETT. 
 
 * 
 
 . 
 
 INTRODUCTION. 
 
 Those planets whose mean daily motions are approximately 600" are classed with the 
 planet Hecuba, or, in the group for which 
 
 u= = K1-0 
 n 2 
 
 where n' and n are the mean daily motions of Jupiter and the planet, respectively, and w is a 
 small quantity. 
 
 Among the minor planets discovered by James C. Watson there are several of this type. 
 In the course of the general program of determining the perturbations of the Watson asteroids, 
 there arose the necessity of computing special tables for the Hecuba group in preparation for 
 the application of Bohlin's method to individual planets. 
 
 General tables for the group $ were in the process of construction, under the direction of 
 Professor Leuschner, 1 according to the method of Bohlin,* when tables for this group were 
 published by H. v. Zeipel.* The computers, Dr. Sidney D. Townley and Miss Adelaide M. Hobe, 
 made a comparison of their tables with those of v. Zeipel and found certain discrepancies 
 Because of this fact the completion of the tables for the Hecuba group was deferred. These 
 discrepancies have been explained, as a result of a careful investigation, and the tables have 
 been completed by Miss A. Estelle Glancy and Miss Sophia H. Levy, under the direction of 
 Professor Leuschner. 
 
 In the completion of the tables, v. Zeipel's method and order of procedure have gener- 
 ally been followed. There are numerous discrepancies between our tables and v. Zeipel's. As 
 far as possible, with the aid of the original manuscript, kindly forwarded by the author, we have 
 traced the source of these disagreements. In some of the more complicated functions it was 
 not possible to do so, and these discrepancies remain unexplained. Our own results, however, 
 are substantiated by the employment of independent developments. Further, where we found 
 terms omitted which were of the same order as those which were included, we frequently 
 extended the tables. In this connection, it is pertinent to remark that it is very difficult to set 
 up a consistent criterion for the omission of terms. With the exception of a few scattered 
 negligible terms, our tables are published in full. They contain terms which may ordinarily be 
 omitted, yet their numerical magnitudes depend upon the elements of the particular planet 
 under consideration, and their use is left to the computer's judgment. Many of them are 
 incomplete, i. e., the tabulated coefficients do not necessarily include all the terms of a given 
 degree in the eccentricities or mutual inclination or of the small quantity w, which depends 
 upon the difference between the planet's and twice Jupiter's mean motion. In other words, the 
 coefficients may not contain all the terms of a given degree having the factors 
 
 W, Jf, ,', ft 
 
 which are defined on page 12. But, assuming certain numerical limits for the fundamental 
 auxiliary functions, the coefficients are of this magnitude. The value of the additional terms 
 will be shown best in an application of our tables to the same planet for which v. Zeipel computed 
 the perturbations. 
 
 Unless stated otherwise, the references to Bohlin refer to the French edition and are 
 designated by B. References to v. Zeipel are designated by Z. 
 
 1 Memoirs of the National Academy of Sciences, Vol. X, Seventh Memoir, p. 200. 
 
 Fonneln und Tafeln rur gruppenweisen Berechnung der allgemeinen StSrungen benachbarter Planeten (Tpsala, 1896). 
 
 Sur le DeYetoppement des Perturbations Plangtaires (Stockholm, 1902). 
 1 Angenaherte Jupiterstorungen fflr die Hecuba-Gruppe (St. Pfitersbourg, 1902). 
 
 9 
 
tw 
 
 I. FORMULAE AND TABLES FOR THE HECUBA GROUP, ACCORDING TO THE 
 THEORY OF BOHLIN-v. ZEIPEL, AND AN EXAMPLE OF THEIR USE. 
 
 DETERMINATION OF CONSTANT ELEMENTS AND OF PERTURBATIONS OF THE MEAN ANOMALY. 
 
 The planet (10) Hygiea was selected by v. Zeipel as an example of the use of his tables for 
 the group . We have used it as a preliminary example for the application of our own tables, 
 so as to provide further comparison of our tables with those of v. Zeipel. 
 
 This example is presented with the direct purpose of meeting the needs of the computer. 
 For this reason, no attempt is made to explain the significance of the functions involved, yet 
 their use will be less mechanical, if, in a general way, some of the essential principles under- 
 lying their development are understood. The theory of v. Zeipel is taken up in the second 
 section of this memoir. 
 
 The method proposed by v. Zeipel is a practical adaptation of Bohlin's method of com- 
 puting the perturbations by Jupiter upon planets whose mean motions bear nearly commen- 
 surable ratios to that of Jupiter. In particular, the formulae are derived for the planets of the 
 Hecuba group. Tracing the history of this method one step further back, Bohlin's method is a 
 modification of the theory of Hansen for the indeterminate case of nearly commensurable mean 
 motions. Or, concisely, in v. Zeipel's own words, "Die benutzte Methode kann einfach dadurch 
 charakterisirt werden, dass die Differentialgleichungen von Hansen mittels des Integrations- 
 verfahrens des Herrn K. Bohlin gelost worden sind." 1 
 
 Certain principles of Hansen are fundamental to an understanding of some of the important 
 equations. Briefly, the perturbations are reckoned in the plane of the orbit and perpendicular 
 to it. In the plane of the orbit n5z signifies the displacement in the planet's mean anomaly 
 (8z is the perturbation in the time) ; v gives the disturbed radius vector through the relation 
 
 u 
 and the displacement in the third coordinate is denoted by =. With Hanson's choice of ideal 
 
 COS v 
 
 coordinates, the fundamental analytical relations are: 
 
 t nl 
 
 s e sin s = nt + c + ndz 
 
 rcosf=a (cos e -) ft) 
 
 fcH!?i /r 3_:_ 
 
 > in/_ * in 
 
 - 
 
 s<( vlhji' 
 
 j'jn 
 
 oiii ni'iftl 1-J vuul . I.HP 
 
 ^ = coin s * n *" 
 
 Jz=o"/cosa (2) 
 
 Jv=/8co b 
 
 "<*' / Jo 
 
 Az = dp cos c 
 
 x = r sin a sin (A' +/) +Ax 
 
 y = r sin b sin (B' +/) + Ay (3) 
 
 z = r sin c sin (C' +/) + Az 
 
 where s,f, f are fictitiously disturbed coordinates, which, in connection with constant elements 
 and the perturbations n5z, v, and = give the true position of the body. A', B', C', sin a, sin b, 
 
 COS 1 
 
 sin c are the constants for the equator. The notation for the eccentric anomaly and the true 
 anomaly is v. Zeipel's; in Hansen's notation they would be written e,f. 
 
 ' Angenaherte Jupiterst8rungen fur die Hecuba-Gruppe, p. I. 
 10 
 
NO. 8.] MINOR PLANETS LEUSCHNER, GLANCY, LEVY. 11 
 
 When Jupiter's mean motion and that of the planet are nearly commensurable, the inte- 
 gration of Hansen's differential equations becomes impracticable through the presence of large 
 integrating factors. The integrals are of the form: 
 
 1 sinf 1 oo <i< + 00 
 
 \(in-i'n')t\ A ^.,^ 
 cos 0<t'<+oo 
 
 ./. i'n'Y 
 
 71*1 1 I 
 
 V n / 
 
 For the Hecuba group the mean motion is approximately twice the mean motion of Jupiter. 
 Hence, for exact commensurability, 
 
 n2' V. iVV 
 nH % -- 
 V n ) 
 
 By introducing the exponential in place of the sine and cosine, the indeterminateness can 
 be removed, for if in i'n' = Q, then V-Km-i'')<_ j This is one of Bohlin's modifications. 
 
 For any given planet the ratio is not exactly commensurable, and the developments are 
 originally made for the case of exact commensurability. They are then expressed, for a given 
 case, by Taylor's series in ascending powers of a small quantity w, which depends upon the 
 difference between the real ratio and exact commensurability. In addition to positive powers 
 of w there will occur negative powers. They are due to the following causes. An argument 6 is 
 introduced (see p. 13), from which the mean anomaly of Jupiter is eliminated through the intro- 
 duction of w. It is a necessary consequence of the form of the partial differential equations in 
 
 j r\ 
 
 which -r appears, that the integration of first-order terms shall contain vr l and that higher 
 
 order terms shall contain other negative powers. Hence the integrals are series in both posi- 
 tive and negative powers of w. 
 
 In distinction to the method of Hansen the elements appear explicitly in the arguments 
 or as factors in the terms of the series. 
 
 An important feature of v. Zeipel's theory is his treatment of the constants of integration. 
 Since the method is essentially Hansen's, the constants of integration must be determined con- 
 sistently with that method. Given osculating elements, the constants of integration are deter- 
 mined by the condition that, at the date of osculation, (t = 0), the perturbations and their 
 velocities shall be equal to zero. 
 
 v. Zeipel adopts osculating elements as his initial elements. With these elements and the 
 perturbations and their velocities at the date of osculation, he computes elements, designated 
 by the subscript unity, in which the constants of integration are absorbed. They are analogous 
 to Hansen's constant elements, i. e., the fundamental equations of Hansen are valid. 
 
 Our transformations of the elements differ from v. Zeipel's in two respects. First, the 
 
 constants in - ., and in its velocitv have not been introduced into the elements i, Q, but 
 cos ^ 
 
 are treated in the usual Hansen manner. Second, v. Zeipel introduces certain terms in the 
 perturbations which have the same period as the planet (argument s), into the elements to 
 form mean elements. This has not been done. 
 
 The general tables, XXXV, XXXVHI, XLIII, LIV, LVi, LVn, LVI, LVTT, which are 
 required in computing the perturbations, are given at the conclusion of the formulae. The 
 formulae for any planet of the group are given completely, and they are supplemented by 
 numerical values for the planet (10) Hygiea. 
 
 The references to v. Zeipel's paper are indicated briefly by Z, followed by the number of 
 the page. 
 
 The osculating elements of the planet are taken from Z 139; the elements for Jupiter are 
 taken from Astronomical Papers of the United States Xautical Almanac Office, Vol. VII, p. 23. 
 
12 
 
 MEMOIRS NATIONAL ACADEMY OF SCIENCES. 
 
 tvoi.xiv. 
 
 (10) Hygiea. 
 Epoch, 1851, Sept. 17.0, Ber. M. T. 
 
 OSCULATING ELEMENTS. 
 
 7i = 634?850 = 0? 176347 
 *> = 546/28= 5?7713 
 ;r = 227 46.61=227.7768 
 ft = 287 37.19 = 287.6198 
 = 300 9.42=300.1570 
 i n = 3 47.14= 3.7857 
 
 Jupiter. 
 Epoch, 1851, Sept. 17.0, Ber. M. T. 
 
 'illl ',' ', r. -\\ L --in: 
 
 MEAN ELEMENTS. 
 
 n'= 299?1284= 0?0830912 
 
 <p' = 245/95= 2?7658 
 
 JT'~ 11 54.45= 11.9075 
 
 ft'= 98 55.97= 98.9328 
 
 ' = 272 58.48 = 272.9747 
 
 i'= 1 18.70= 1.3117 
 
 to*-! 
 . o 
 
 . C = 126 59.81 = 126. 9968 c' = 199 57.70= 199. 9617 
 
 Mean equinox and ecliptic, 1850.0. 
 Epoch, 1851, Sept. 16.96279 Gr. M. T. 
 
 The following notes in regard to these elements are of importance: 
 
 Jupiter's elements were first taken from Z 139. They were used only in the equations 
 numbered (1). In these equations either set of elements may be used with sufficient accuracy. 
 In fact, it is not necessary to know Jupiter's elements as accurately as those of the planet, for 
 they appear only in the arguments of the perturbations. We have adopted Hill's values of 
 the elements and Newcomb's value of the mass of Jupiter. The tables of the perturbations 
 are based, however, on Bessel's value for m'. To correct the perturbations for the improved 
 value, it is only necessary to multiply them by 1.0005, and this is done in the formulae which 
 follow. 
 
 The original epoch of Jupiter's elements was 1850.0 Gr. M. T. It was changed by the 
 
 formula c' = 148 1/97 + '* (4) 
 
 '<}<; iv (!.; < 
 The elements of Hygiea are very good osculating elements, computed by Zech. They 
 
 include perturbations by Jupiter, Saturn, and Mars and are based on five oppositions. The 
 reference for these elements is doubtful, for in Astronomische Nachrichten 39, 347, the elements 
 given by Zech are not identically the same, although the differences are very small. The values 
 given by v. Zeipel were probably taken from Zech's manuscript, to which he had access. They 
 may, therefore, contain some later corrections. 
 
 The auxiliary quantities ^, *, J are first computed by the formulae: 
 sin ^ J sin ^ (*+*)=sin^ (ft - ft') sin i (i +*') 
 
 sin ^ J cos g (^+*)=cos ^ (ft - ft') sin 2 (*o~*') 
 cos K J sin o (*-*)= sin ^ (ft -ft') cos 7, (i +i') 
 
 (5) 
 
 Then follow 
 
 cos sr J cos K ( 1 i r -*)= 
 
 sin Sfr sin $ sin 
 
 
 , &') cos o (*o~ *') 
 
 sin i sin i' 
 
 
 
 - cos 
 
 
 cos 2 
 
 
 s- 
 
 J =n -n' ; j- = 
 
 cos 
 
NO. a.] MINOR PLANETS LEUSCHNER, GLANCY, LEVY. 13 
 
 and the arguments for the date of osculation: 
 
 .nwbl-M 
 
 o = Lr - g' where g' = f ' + [n'fel ; [n'te'J = (9.5215) sin. 1 15?326, (7) 
 
 where the coefficient in parentheses is logarithmic in degrees. 
 
 1 
 e e sme, = c :r = ^e <> + + J lt (8) 
 
 (10) Hygiea. 
 
 * = 186?4792 n, = 302?3984 J. = 215?8679 
 
 * = 357. 7586 n' = 86.5305 J.= 28.9289 
 J= 5.0856 
 
 log ,= 8.70139 log /* = 7.29275 0, = 223.2334 (a) 
 
 log i)' = 8.38238 log i = 8.94739 ^ e ~ ( c ' + f 72 '^^ = 223 -2445 (6) 
 
 = 8.76072 \ c,-c' = 223.5448 (c) 
 
 See footnote. 1 
 e = 131?3236; r=145?0746 
 
 With these initial quantities all the arguments and factors in Table L^*I or F are computed. 
 The required function, w w t , is computed by successive approximations, the first approximation 
 being 
 
 Wg 
 
 In the first trial the smallest terms and the last digit may be omitted; the second trial should 
 be accurate; a third trial, if necessary, will require only corrections to the largest terms. 
 The mean motion n is then given by 
 
 In' 
 n== l^i> 
 
 honinmlob -i ., Td bsloiwb 0=-- \ -..u) - * -.ui.,,>. badiuJ.ii. 7fcooiJii-.il 9ii 
 
 -noc Jam id* jbuotuj 
 
 (10) flj/yiec, 
 
 _ ,, . . , , 
 
 The three successive trials for u> give 
 
 W -w. 
 
 + 0.00388 w = + 0.06 1 208 
 
 + 0.003541 logw= 8.78681 
 + 0.003568 n = 637?2633 
 
 . 
 
 Designating by C and S series to be computed next from Table LVII or G, it is evident by 
 inspection of Table LYII that 
 
 C'cos i^ + S sin <}> = Ic cos (i{> + X)=Ic cos X cos <pZc sin X sin $ 
 from which 
 
 C= Ic cos X; S=-Ic sin X (10) 
 
 > Three numerical values for the argument i, are given. According to the theory (see footnote, Part 2, p. 147), (a) is rigid; (4) is rigid within the 
 accuracy of the developments by v. Zeipel; (e) is an approximation which v. Zcipel used and which is used here. The value (i) is preferable. 
 
 Inequation (6), [n'Sz 1 ] +0*.31U and is tho complete perturbation of Jupiter by Saturn taken from Hill; in all other parts of the computation 
 n'li'} is only the long period term used by v. Zeipel. 
 
14 
 
 MEMOIRS NATIONAL ACADEMY OF SCIENCES. 
 
 [Vol. XIV. 
 
 To make the order of computation evident, the successive steps for a group of terms for 
 Hygiea are given. 
 
 X 
 
 c 
 
 X 
 
 -s 
 
 + C 
 
 -5r+60 +6J 
 
 + 0.4 
 
 * ' 
 
 111. 10 
 
 + 0.4 
 
 ft 
 
 - 0.1 
 
 -4r+60 +64, 
 
 + 1.9 
 
 256. 18 
 
 - 1.8 
 
 - 0.5 
 
 -3r+60 +6J 
 
 + 4.6 
 
 41.25 
 
 + 3.0 
 
 + 3.5 
 
 -2r+69 + 64, 
 
 + 6.8 
 
 186. 33 
 
 - 0.7 
 
 -6.8 
 
 - r+60 +64, 
 
 +21.5 
 
 331. 40 
 
 -10.3 
 
 +18.8 
 
 60 +6J 
 
 -63.0 
 
 116. 48 
 
 -56.4 
 
 +28.1 
 
 r+60 +6J 
 
 - 4.0 
 
 261. 55 
 
 + 4.0 
 
 + 0.6 
 
 2r+60 +64 
 
 - 3.1 
 
 46.63 
 
 -2.2 
 
 - 2.1 
 
 3r+60 +6J 
 
 - 1.9 
 
 191. 70 
 
 + 0.4 
 
 + 1.9 
 
 
 The second column contains the sum of the numerical coefficients multiplied by their respec- 
 tive factors i&ipy'vf 1 \ The columns S and + C con tain the required terms from this group in 
 the table. They can be computed at the same time if a traverse table is used. 1 
 
 From S and C the elements n and <p can be computed by the formulae : 
 
 i 
 
 e sin (n 7r )=S cos <p u 
 
 e cos (n TT O ) = e + C cos V 
 
 e = sin <p 
 
 In place of ij a , J , - the following are used hereafter: 
 e 
 
 -TOOT-* 
 
 
 (ID 
 
 (ff TO) 
 
 .dj 
 
 (12) 
 
 (70) 
 
 
 S=+1215?0 
 <7= +2191.1 
 
 A = 218?8882 
 
 ;:= +3?0203 
 r = 230. 7971 
 S= 31?9492 
 
 log, = 8.7451 7 
 
 IB hi 1-ift >r!t rtl 
 
 (I ft I'.KtBIU'i'MI 9<i 
 
 (io(, bdT 
 
 There remains one more element to determine, namely, c, but the computation must be 
 deferred until we know the perturbation nSz at t = 0. (See equation (1), page 10 or page 16.) 
 
 The fictitiously disturbed eccentric anomaly at the time t = denoted by is determined 
 through the relations : 
 
 (13) 
 
 s n e n sin 
 
 where is calculated with the aid of Astrand's table 2 ; 
 
 iA rlivilj *tVMSi9*V 
 
 (14) 
 
 v<i 
 
 131?3236 
 li.Oio TIV.I 'iki 
 
 (10) Hygiea. 
 
 ! = 127?6064 
 
 i-csine, = 122?5578 
 
 The perturbation nSz is computed as follows : 
 
 The function 1 + 0(0) is computed from Table XXXVIII or B. The coefficients are mul- 
 tiplied by their respective factors, the trigonometric functions of the arguments are expanded, 
 
 and the coefficients of 
 
 5111 
 
 are collected, (j is the numerical coefficient of t?) . 
 
 1 Memoirs of the National Academy of Sciences, Vol. X, Seventh Memoir, p. 218. 
 
 Hulfstafeln zur leichten und genauen Auflosung des Kepler'schen Problems (Leipzig, 1890). 
 
Ko.3.) 
 
 MINOR PLANETS LEUSCHNER, GLANCY, LEVY. 
 
 15 
 
 (10) Hygiea. 
 
 1 + 0(0) = (1 -0.008064) {1 -0.055937 sin 20 + 0.017170 cos 20 
 
 + 0.016057 sin 40 + 0.012244 cos 40 
 + 0.000905 sin 60-0.005081 cos 60+ ..... 
 + (*-*.)( + 0.000007 -0.000490 sin 20-0.001266 cos 20 
 
 -0.000361 sin 40 + 0.000409 cos 40+ ..... )} 
 
 where the coefficients are in radians, and is the value of at t = 0. 
 
 
 -j ji iwjilq *Jii) Wi .-f.R ' ,-. PI imirfJiiHSpI 'y:ii ,j, xitulv, ;., 
 
 Let 1 + a be the nontrigonometrical term in 1 + 0(0), take it out as a common factor, and 
 denote the numerical coefficients by A 2 , B v A v B v A t , B s , b a , a 2 , b 2 , a t , b t , respectively. 
 With these coefficients the following are computed : 
 
 K= T^w sin 1" 
 
 
 
 
 
 (15) 
 
 r> 
 
 C. 
 
 Ift'.V I'.. 
 
 There are check formulae for these quantities in Z 134, equation (153), (161')- In equa- 
 tion (153) there is a misprint; in equation (161') there are two misprints. The errors and their 
 corrections are noted in the list of errata which accompanies the second section of this paper. 
 
 A part of the long period terms in ndz, denoted by [ruJz],, is expressed by 
 
 sn 
 
 cos 
 
 sn 
 
 cos 
 
 sn 
 
 cos 
 
 ^^ 
 
 
 (10) Hygiea. 
 
 1+0(0) = (1-0.008064) {1-0.056384 sin 20 + 0.017308 cos 20 + 0.016186 sin 40 
 
 + 0.012342 cos 40 + 0.000912 sin 60-0.005122 cos 60+ . . . 
 + (0-0,) ( + 0.000007 -0.000494 sin 20-0.001276 cos 20-0.000364 sin 40 
 
 
 
 + 0.000412 cos 40+ 
 
 A, = + 0.01 7308 
 Bj = - 0.056384 
 AI= +0.012342 
 B,= +0.016186 
 A t =- 0.005 122 
 =+ 0.000912 
 
 ' 
 
 . . )+....} 
 6, = +0.000007 
 a, = - 0.000494 
 6, = - 0.001276 
 a 4 = -0.000364 
 6 4 = +0.000412 
 
 
 
 
 
 
 \ Ho'Ji nc:ij'u:j' 
 
16 MEMOIKS NATIONAL ACADEMY OF SCIENCES. [VOLXIV. 
 
 Unit of A 2 , etc., is one radian 
 
 [<H= (3.59592) sin 2 +^(C~Co) [(0.933J sin 2 
 + (4.09785) cos 2 + (0.521) cos 2 
 
 + (3.0783) sin4 +(0.085) sin 4^ 
 
 + (3.2230J cos 4^ + (0.005) cos 4 
 
 + (2.4390.) sin 6C + ...... ] 
 
 + (1.494,,) cos 6C + v ,w**i (! 
 
 in which the coefficients are logarithmic in seconds of arc. For this planet it is not necessary 
 to include C ". 
 
 uu mis t 
 
 In equation (16) let 
 
 S n = fc cos K C n = sin A . 
 
 S' n = -fc' sin J5T' C",-*' cos JT 
 
 Then 
 
 cos (c+^')+ Jv t u.-, (18) 
 
 The argument C ia given by the relation: 
 
 (51) 
 
 and ^ is the value of at 2 = 0, in which, [w'fe'J, the long period term between Jupiter and 
 Saturn is: 
 
 
 
 (9.5215) sin{ (9.58539) T+ 1 15?326}, R) (20) 
 
 where the numerical coefficients are logarithmic in degrees, and Tis measured from the date of 
 osculation in Julian years. 
 
 The complete expression for the long period term in ndz is : 
 
 ;?.!) - 2 o-KAS + B^/w ,,\ wriT 
 
 - [ndsll+ l-wl + $(A 2 2 +B 2 2 )\2 s [n ' zl j 
 
 It is important to remark that, in equations (19), (21), the eccentric anomaly is computed 
 by the usual formula, 
 
 ' JO 4- } iu* s ~ e sin $ = c + nt + n8z : wo ?> - Jata] C 1 ) 
 
 in which the multiples of 2^ must be retained, for is used here as if it were the time. Since 
 ndz is unknown, the computation is by successive approximations. 
 
 (10) Hygiea. 
 
 [7^3],= (4.1 1837) sin (2+ 72?5246) 
 +(3.3130) sin (4C + 305. 627) 
 + (2.442) sin (6C + 186.48) 
 -.o'jpV-'JUO.O ..... i'lH )( iJ)jl).-.TO 
 
 [ +|(C-Co){(0-963)cos(2C+ 68.83) 
 
 + (0.199) cos (4C+309.75)+ ....}+ .... 
 in which the coefficients are logarithmic in seconds of arc. 
 
 lQ g i 
 
 The argument & in (ndz [ndz]), the short period part of 71^2, is given by 
 
 C (22) 
 
 and the function itself is computed from Table XXXV or A. 
 
NO. 3.] MINOR PLANETS LEUSCHNER, GLANCY, LEVY. 17 
 
 The numerical coefficients in Table XXXV or A are multiplied by their respective factors 
 
 and the terms are then collected in the form 
 
 r^z-[ndz] = lC S ^(i^e+j^ + U-H) (23) 
 
 By expanding the trigonometric functions, the known part of the argument, namely, 
 
 IcA-lZ 
 
 is incorporated in the coefficients, and the terms are collected in the form : 
 ndz - [ndz] = la sin x + 2b cos 
 
 where 
 
 x=2 
 Let 
 
 a = cos K 
 a' =- sin K' 
 a"= Jc" cos K" b" = Jc"smK" 
 
 Then 
 
 ndz - [ndz] = Ik sin Gt+Z) + (*-*,) SV cos (x+ K' ) 
 
 >-tfo 
 
 mo 
 
 ' sinx + -^' cos x) 
 " sin x + 2b" cos x) 
 
 (24) 
 
 6 
 b' 
 
 =t 
 =&' 
 
 sin K 
 cos 1C' 
 
 (25) 
 (26) 
 
 The tabulation of ndz [ndz] for (10) Hygiea is given on page 27. 
 
 Finally, the complete perturbation in the mean anomaly is: 
 8 ndz = [ndz]+(ndz-[ndz]) (28) 
 
 It is now possible to determine c by successive approximations from equations (20), (19), 
 (18), (21), (22), (27), (28). 
 
 From equation (1), which holds for any time t, 
 
 c=ie sin ^ ndz 
 
 t = o (29) 
 
 = i 
 As a first approximation 
 
 ndz = c = e, e sin s l 
 
 Introducing this value of c in equation (19), a first approximation for ndz is made. For <=0, 
 
 C-Co)=0 (30) 
 
 (*-0 8 ) =0 
 
 Substituting the value of ndz in equation (29), and computing a new value of c, the process of 
 solution by trials is repeated until a satisfactory agreement is reached. 
 110379 22 2 
 
 -K?;J 
 
18 
 
 MEMOIRS NATIONAL ACADEMY OF SCIENCES. 
 
 [Vol. XIV. 
 
 (10) Hygiea. 
 Below is the last approximation for the constant c. 
 
 (See tabulation of mz \n3z\ on pg. 27.) 
 
 L'l. 
 
 
 
 24** 
 
 x+K 
 
 log sin (*+ A') 
 
 , Si n (x+ .) 
 
 Approx. ndz 
 
 +0?6124 
 
 
 
 
 
 
 [ sin , 
 
 122. 5578 
 
 i*+ & 
 
 285?683 
 
 323?619 
 
 9. 7732 n 
 
 - 282" 
 
 Approx. c, equ. (1), p. 10 
 
 1 w 10 
 -g- c > P- 13 
 
 121. 9454 
 57. 240 
 
 frS! 
 
 9.443 
 93. 203 
 
 291. 021 
 258. 21 
 
 9. 9701 B 
 9. 991 n 
 
 - 680 
 - 260 
 
 *H2! r r 7 n 1 
 
 217. 278 
 
 lff+30 
 
 158. 077 
 241. 837 
 
 183.00 
 335. 37 
 
 8. 719 n 
 9. 620 n 
 
 - 6 
 
 - 17 
 
 o C (, , p. -L* 
 
 
 
 
 
 127. 606 
 
 135. 14 
 
 9.848 
 
 + 25" 
 
 I.UP-W 
 
 +3. 9053 
 
 +20 
 
 211. 366 
 295. 126 
 
 288. 414 
 256.179 
 
 9. 9772 n 
 9. 9872 n 
 
 -3403 
 - 723 
 
 [n'oV], equ. (20), p. 16 
 
 +0. 3003 
 
 + 60 
 
 18. 886 
 
 223. 38 
 
 9. 837 n 
 
 - 168 
 
 (9.99572)(|f 1 -K<?2']) 
 
 +3. 5697 
 
 +80 
 
 102. 646 
 316. 154 
 
 186.8 
 14.11 
 
 9. 073 n 
 9. 387 
 
 - 5 
 + 23 
 
 
 
 - +40 
 
 39. 914 
 
 129. 91 
 
 9.885 
 
 + 3 
 
 ( \ 
 
 
 
 137. 049 
 
 74.51 
 
 9.984 
 
 + 121 
 
 5 i [n'dz'] j 
 
 -0. 0752 
 
 ff+50 
 
 220. 809 
 
 39.53 
 
 9.804 
 
 + 44 
 
 ' 
 
 
 
 304. 569 
 
 3.25 
 
 8.754 
 
 + 1 
 
 f, equ. (19), p. 16 
 
 220. 848 
 81. 696 
 
 2f+40 
 
 338. 972 
 62. 732 
 
 236. 180 
 209. 15 
 
 9. 9195 n 
 9. 688 n 
 
 - 80 
 - 19 
 
 , 
 
 163. 392 
 
 2e+60 
 
 146. 492 
 
 183.0 
 
 8. 72 n 
 
 - 1 
 
 fir 
 
 245. 088 
 
 s +50 
 
 348. 415 
 
 2.4 
 
 8.62 
 
 
 
 
 
 tt+n 
 
 72. 175 
 
 327.9 
 
 9.72 n 
 
 - 4 
 
 2+ 72?525, p. 16 
 4r+305. 627 
 
 154. 221 
 109. 019 
 
 
 
 
 
 + 217"-5654" 
 
 6r+186. 48 
 
 71.57 
 
 
 
 
 ndz [noz] 
 
 / - 5437" 
 {- 1?5103 
 
 log sin 
 log sin 
 log sin 
 
 9. 6384 
 9. 9756 
 9. 9771 
 
 
 
 
 (8.3192 B )(| 1 -KoV]) 
 
 [nM 
 
 7102, equ. (21) 
 
 - 0.0752 
 
 + 2. 1994 
 + 0.6139 
 
 
 + 5712" 
 
 
 
 
 C=C 1 
 
 121. 9439 
 
 
 + 1944 
 
 
 
 
 
 
 
 + 262 
 
 
 
 
 (6. 8050 B V , 
 
 ^ 1 ! 1 
 
 - 0.0778 
 
 me* 'i i 
 
 
 / +7918" 
 
 
 
 
 c.,, p. 19 
 
 . ODD! 
 
 (9. 67154)^2], 
 
 \ +2? 1994 
 +1. 032 
 
 
 
 
 l-w 
 
 +57. 240 
 
 2 C> 
 
 
 
 
 
 
 lw 
 
 
 0, equ. (22), p. 16 
 
 221?880 
 
 
 
 
 
 217. 278 
 
 2 q c f 
 
 20 
 
 83.760 
 
 
 
 
 
 
 
 305. 640 
 
 
 
 
 (9. 6715)1^2], 
 
 + 1.032 
 
 .- - ; -.' 
 
 
 
 
 
 , equ. (22) 
 
 221. 880 
 
 40 
 
 167. 520 
 
 
 
 
 
 
 50 
 
 29.400 
 
 
 
 
 
 
 60 
 
 251. 280 
 
 
 
 
 
 
 70 
 
 113. 160 
 
 
 
 
 
 
 80 
 
 335. 040 
 
 
 
 
 
 
 Jf , p. 14 
 
 63.803 
 
 
 
 
 
 
 f 
 
 127. 606 
 
 
 
 
 
 
 t, 
 
 191. 409 
 
 
 
 
 i i I If tl) I 
 
 
 2f 
 
 255. 212 
 
 
 
 
 
 
 
 319. 015 
 
 
 
 
 
 
 
 
 
 
 i 
 
 Collecting the elements, and adopting a change of notation, introduced at this point by 
 v. Zeipel, namely, the addition of the subscript unity to the elements just now determined, 
 
 n, = 637?2633 = 0? 17701 758 
 <?,= 6? 3858 
 ^ = 230.7971 
 c, = 121. 9439 
 
NO. 3.] MINOR PLANETS LEUSCHNER, GLANCY, LEVY. 19 
 
 These elements are constants; they differ from constant osculating elements only by the 
 constants of integration in ndz and v. They are to be used in the same manner as Hansen uses 
 constant osculating elements. 
 
 It is possible, in a similar manner, to absorb the constants of integration in the third coordi- 
 nate in the elements i and &, but this transformation will be omitted. 
 
 It is a convenience to the computer to have n, and c t transformed to mean elements. The 
 last term in equation (21) increases in magnitude, progressively with the time. The computa- 
 tion of this term of large magnitude may be avoided by modifications of the elements n, and c,. 
 
 The method of transformation can be clearly shown from the example (10) Hygiea, 
 
 T^wfl l(l**+^ > )(?*~^ n/te ^) = ( 6 - 80497 ) + (8.3192) [n'dz'] (31) 
 
 By equation (1) 
 
 (6.80497,)E + (8.3192) [n'dz'] = (6.80497 ,) c, - Of 4067 t - 14f6 sin E 
 + (6.80497,)n<fcr + (8.3192)[/<Jz / ] 
 
 It is evident from equations (1), (21), and (23) that the first term on the right-hand side 
 of equation (32) may be combined with the mean anomaly at the epoch to form a mean mean 
 anomaly, given by Cj _ ^ + (6 .80497 Jc, 
 
 Furthermore, the second term on the right-hand side of equation (32) may be combined 
 with nt in equation (1). A mean mean motion is thereby introduced, which is given by 
 
 n, - n, - Of 4067 = 636f8566 
 
 Again, the third term on the right-hand side of equation (32) may be combined with a 
 term in (ndz [ndz]) which has the argument E. In the construction of (ndz [ndz]) there 
 occurred the terms + 34fg gin +4 , 6 co = (1 545) ^ ( + 7 o 53) 
 
 The addition of 14f6 sin E from equation (32) gives 
 
 + 20f2 sin E+4f6 cos E = (1.320) sin (e+12?74) 
 
 These two values for the argument x = E are tabulated in the body of the table given on p. 27. 
 
 Further, since it is intended to improve the perturbations by the use of Xewcomb's value 
 for the mass of Jupiter, ndz must be multiplied by the factor 1.00050. The combination of the 
 correction for the mass of Jupiter with the term of the same form in equation (32) gives 
 
 ( + 0.00050 -0.00064)7nJ2 = -0.00014 ndz 
 
 This correction is the last step in the determination of ndz, since it depends upon the pertur- 
 bation itself. 
 
 Without change of notation for ndz, the collected results are: 
 
 E e sin E = c 2 + ndz + nj (33) 
 
 where 
 
 ndz = [ndz\ + (ndz - [ndz]) - 0.00014 ndz + (8.319) [n'dz'] (34) 
 
 It must be remembered that [ndz] t and (ndz [ndz]) are numerically different from their 
 original values, but there should be no confusion if this transformation is not made before the 
 constant c has been determined. 
 
 The constant elements are now: 
 
 Epoch and Osculation, 1851, Sept. 17.0, Ber. M. T. 
 
 n, = 636f 8566 = 0? 17690461 
 c, = 121?8661 
 <P l = 6.3858 
 *, = 230.7971 
 ft = 2S7.6198 
 ' = 3.7857 
 
20 MEMOIRS NATIONAL ACADEMY OF SCIENCES. 
 
 Equinox and ecliptic, 1850.0 
 
 
 logP = 9.98741 log ^ = 9.04620 log p 2 = 0.49191 
 
 X= 9.95150 
 
 V-i _ 
 fX7= 9. 
 log a* = 1.491 93 log a t = 0.49731 
 
 Certain other transformations of the elements which v. Zeipel makes are omitted. Those 
 terms of the perturbations which have the argument s have the same period as the planet and 
 can, therefore, be absorbed in the elements. It would be necessary to set up formulae for this 
 transformation to mean elements, and it is not profitable to do so. 
 
 PERTURBATIONS OF THE RADIUS VECTOR. 
 
 The perturbations in the radius vector are computed in a manner similar to that for 
 (nds [nUz]). In Table XLIII the numerical coefficients are multiplied by their respective 
 factors w*, -if, 9'', J*, the terms are collected, the known parts of the arguments are incorporated 
 in the coefficients, and the terms are grouped in the form: 
 
 v i vl 
 
 v = 2 a sin x + 2o cos \+ 
 
 + (tf-#o) ( 2a> sinx+JZ>' COSX+- } (35) 
 
 -K#-tM J Ua" sin x + 2-fc" cos x+ !*.!;!.} + 
 Let 
 
 a = It sin K I =Jc cos K 
 
 a' = Jc' cos K' b' =' sin K' (36) 
 
 a"= -V sin K" &"=-*" cos K" 
 
 Then 
 
 v = 2Jccoa (x+ft + W-flJZk'smb+IO + W-WWcos ( X +K") + - (37) 
 
 and to correct the perturbation for the use of the improved value of the mass, i> should be 
 multiplied by 1.00050. 
 
 If the mean motion n t is adopted, the constant in v must be corrected by 
 
 t 1 
 
 3 n t sin 1" 
 This correction of the constant in v permits the use of the relation 
 
 7i 3 2 a 2 3 = P 
 
 in the computation of a geocentric place; without this correction it would be necessary to use 
 the relation 
 
 nfaf = P 
 
 in the determination of the parameter p. In the computation of the eccentric anomaly it 
 is permissible to use either n l or n 2 , for the difference is taken up in the modification of 77^2, 
 but the theory of Hansen demands the use of constant elements. Hence, strictly speaking, 
 7i, must be used in computing a geocentric place. The modification of the constant in v renders 
 the employment of n 2 equivalent to the use of n t . 
 
 (10) Hygiea. 
 
 2 n,-ro, 1 _2 Of 4067 1 _, 
 
 3 n, sin l"~ 3 637f3 sin 1"" 
 
 The constant in Table XLIII or C'is +47?6. Therefore, the new constant is: 
 +47?6-87f8=-40?2 = (1.604) cos 180?00 
 
 where the coefficient is logarithmic in seconds of arc. 
 The perturbation is tabulated on page 27. 
 
NO. a.] MINOR PLANETS LEUSCHNER, GLANCY, LEVY. 21 
 
 PERTURBATIONS OF THE THIRD COORDINATE. 
 
 The perturbations of the third coordinate are derived from Tables LTV, LVi, LVn or D, 
 E,, E,. The first of these is of the same form as the tables for (nSz [nSz]) and v. After mak- 
 ing analogous transformations and multiplying by the factor i cos i, (i is defined by equation (6)), 
 
 i cos il U p . q 7jPi ) '''siiiA = 2Jcsw(x+K) (39) 
 
 Both Table LVi or E, and Table LVn or E a lead to a single numerical quantity, since all 
 the factors and arguments are known constants. 
 The perturbation u is given by 
 
 = i cos i [2U p . q r)i>T)'i sin A + njt.{K l (cos e eJ + K, sin e} +c t (cos e e,) + c, sin e] (40) 
 
 in which c,, Cj, the constants of integration, have not been determined. 
 The constants c t and c, are determined by Hansen's conditions: 
 
 (41) 
 
 __ . _ III 
 
 dt 
 
 Substituting these relations and equation (39) in equation (40), the determination of c, and 
 c, is given by the solution of 
 
 C l (cosf-e 1 ) + C t sias=-IJcsin(x+K); C t sin e - <7, cos e = 2lc ^ cos (x + K) (42) 
 
 where <7, =- 1 cos i.c, , _. 
 
 C. = i cos i.e. 
 and 
 
 dx 
 
 d e 
 where 
 
 dfi l+<r w - 
 
 dt l+HA'+B, 1 )' 2 
 A double notation is used here, for cos i is the cosine of the inclination of the orbit, and is 
 
 M 
 
 the numerical coefficient of e in the argument x, but this should cause no confusion. 
 Dividing and multiplying the factor 
 
 i cos i-nj, 
 by 365.25 
 
 i cos i-n. rr> 
 1 COS V1 *- -365^5 T (45 > 
 
 where T\s the interval in Julian years, measured from the date of osculation. 
 It is evident that 
 
 C l (cos e ,) + C t sin e 
 can be incorporated in 
 
 Jit sin 
 
 in the same manner as similar terms were treated in (ndz [ndz\). 
 For symmetry of form, let 
 
 c cos i- nj{ KI (cos e-eJ + K, sin e} =2V cos (x + K') (46> 
 
 finally, then, without change of notation, 
 
 M = Jisin (x+ K) + TZk' cos (x+Jf) (47) 
 
 in which the constants of integration are absorbed in the first term. The perturbation u is 
 tabulated on page 27. 
 
 The perturbations in the heliocentric coordinates are computed from equations (3) The 
 signs of cos a, cos b, cos c are determined as follows: 
 
 cosa>OifO<8< 180 
 cos 6<0if -90 <& <+90 
 cos 5 < in any case, if e > i 
 cos c > if sin i cos ft < cos t 
 cos c>0 in any case if i<45 
 
22 MEMOIRS NATIONAL ACADEMY OF SCIENCES. [Voi.xiv. 
 
 (10) Hygiea. 
 
 t = 
 [(4.41940) cos (2Co+ 72?5246) 
 
 + (3.9150) cos (4 
 
 + (3.220) cos (6C + 186?48)]sin 1 
 
 S=- 285 
 
 Jfcsin (x+K)=- 70f5 
 
 <7 
 
 cos (x+K)= + 101f6 
 
 Ci- + 35?9 
 <7,= + 12058 
 
 From Table LIV. multiplied by t cos i we have three terms in 
 
 !-i iioiJoIoa ;flJ.vtf a<v/.^ * ^ 
 
 Jfcsin (x+-K) = -4.'2-lf9sin + 2f7 cos e 
 which, added to 
 
 C",(cos e-ej + C 2 sin s= + 12058 sin + 3559 (cos e-c,) 
 
 gives for two terms in 2" sin (x+ K) 
 
 hfi/i 
 -758 + 11859 sin + 3856 cos =(0.89) sin 270?0+ (2.0970) sin (s + 17?99) 
 
 CHECK COMPUTATION. 
 
 After the elements have been determined and the final tabulation of the perturbations is 
 ready, the following checks should be performed, even if the computation has been duplicated. 
 
 t = 
 
 6 =|(e e sin s)q' 
 
 1 w 
 g' = c' + [n'dz'} 6 = t> + g (ndz - [ndz]) - yw sin 
 
 ?.i.<Vi'. "1 
 
 where the necessary quantities are to be taken from the last approximation for c. 
 Secondly, the heliocentric coordinates 
 
 x-Ax, y-Ay, z-Az 
 
 for t = must check when computed by the usual formulae for two body motion and osculating 
 elements, and when computed with the final set of elements and the corresponding perturba- 
 tions, ndz and v, taken from the final tabulation. 
 
 The final tabulation of the perturbation in the third coordinate is checked by the test 
 
 t=Q u =0 
 
 (8*) 
 
 COMPUTATION OF THE PERTURBATIONS FOR THE TIME t. 
 
 It is well to emphasize here the distinction between the elements n, and c, and the elements 
 n t and c 2 in their relation to the perturbations. Let ndz l denote the perturbation in the mean 
 anomaly computed according to equations (20), (19), (18), (21), (22), (27), (28), and let nz 2 
 signify the perturbation computed according to equations (20), (19), (18), (22), the final tabu- 
 lation of (ndz [ndz]), and an equation analogous to (34). (It must be remembered that equa- 
 tion (34) is for (10) Hygiea only. The numerical coefficients are determined for each planet 
 individually.) 
 
 Before the determination of c there can be no confusion, for there is but one way to com- 
 pute the perturbation ndz. Later, when both c, and c 2 are given, the computation may be per- 
 formed in either manner. The latter method is, of course, adopted. The question then arises, 
 what values of and c are to be used in equation (19) ? 
 
No. 3.] 
 
 MINOR PLANETS LEUSCHNER, GLANCY, LEVY. 
 
 23 
 
 Clearly, there is only one value of e, for 
 
 s i sn 
 
 and both rufe and e must be found by trials. Further, since the introduction of n, and c, arises 
 merely from a transfer of certain terms in the perturbation, the argument of the perturbation 
 is independent of this transformation. Therefore c t is the constant in equation (19). 
 
 For any time t the order of computation is: equation (33), neglecting n9z, (20), (19), 
 (18), (22), final tabulation of ndz [ndz], and the equation analogous to (34). Since the per- 
 turbations are large, the argument is not sufficiently accurate when ndz is neglected. It is, 
 therefore, always necessary to make a second approximation for ndz. In the first trial the small 
 terms may be omitted. 
 
 (10) Hygiea. 
 PERTURBATIONS n8z, v, u, FOR 1873, SEPT. 20.4491, BER. M. T. 
 
 log , (degrees) 
 logf 
 
 0.80432 
 8. 48578 
 
 log sin 
 log sin 
 log sin 
 
 9.9387, 
 9.9918, 
 9.703, 
 
 
 
 1 2 
 
 
 
 
 
 
 
 97*;fil 
 
 
 
 
 
 Iog 57.30 w 
 
 . / tWIi 
 
 
 
 
 
 
 '2177278 
 
 
 
 
 2 
 
 Co 
 
 2207848 
 
 log cos 
 
 9. 741, 
 
 . * 
 
 Vjj 
 
 2217880 
 
 log cos 
 
 9.419 
 
 2 
 
 . 
 
 12178661 
 
 W (r r ) 
 
 1.6337 
 
 
 
 ;*] 
 
 
 2* 
 
 
 4*+ & 
 
 3157360 
 
 o p*>e ' *u 
 
 
 iog^:-: > 
 
 1.3898 
 
 Jf-l~3$ 
 
 119. 524 
 
 
 
 
 
 is -^5^ 
 
 283. 698 
 
 4- ; fli^t *; j < 
 
 1873 
 
 log ^(C-Co) cos 
 
 1.131, 
 
 -+ * 
 
 208.824 
 
 
 
 u 
 
 
 lr_l_3|} 
 
 12.998 
 
 
 
 2 
 
 
 
 
 Ber. M. T. 
 
 Sept. 1 20. 4491 
 
 log-(C-C.)coe 
 
 0.809 
 
 
 
 106. 526 
 
 ( 
 
 + 8039? 4491 
 
 
 
 
 270. 700 
 
 n^l 
 
 + 142272156 
 
 
 11405" 
 
 E+40 
 
 74. 874 
 
 Cj+Hjf 
 
 nz 
 
 1544. 0817 
 104. 0817+1440 
 '3.666 
 
 "A + i A 
 
 2017 
 140 
 124 
 
 +6* 
 + 8t> 
 
 239.048 
 43. 222 
 57 648 
 
 Jf=c,+n,<+na* 
 
 100. 416 
 
 
 +10 
 
 - f+4tf 
 
 221.822 
 
 * 
 
 i ' 106. 526+1440 
 \ 1546. 526 
 
 [*], 
 
 13676" 
 
 |+3a 
 
 61. 876 
 226.050 
 30 224 
 
 log* 
 
 3. 18935 
 
 log [n9z\ (sees) 
 log [nte], (degrees) 
 
 4.13596, 
 0. 57966, 
 
 |+7<> 
 
 194. 398 
 
 log^ 
 
 1. 67513 
 
 [jufc], 
 
 -3. 7989 
 
 -Jt+ t 
 
 102.298 
 
 w 
 
 r 
 
 + 477329 
 
 
 
 2c 
 
 213. 052 
 17.226 
 
 -['*'] 
 
 + 477053 
 
 log (9.6715) [jute], 
 
 0. 2512, 
 
 2+4. 
 
 181. 400 
 345.574 
 
 log fe-KAq) 
 
 1. 67259 
 
 (9.6715) [ndi], 
 
 -17783 
 
 |+W 
 
 136. 750 
 
 * . ' X 
 
 
 
 
 4*+7* 
 
 300.924 
 
 log (9.99572) (J^e- [n'dz'U 
 
 1.66831 
 
 a 
 
 2627087 
 
 
 
 (9.99572)(|e-[n'S^) 
 
 + 467592 
 
 d-0 
 
 407207 
 
 
 
 r 
 
 2637870 
 
 
 
 
 
 
 
 f 
 
 2627087 
 
 
 
 2' 
 
 1677740 
 
 2$ 
 
 164. 174 
 
 
 
 \" 
 
 335. 480 
 
 3i> 
 
 66.261 
 
 
 
 6; 
 
 143. 220 
 
 4t> 
 
 328. 348 
 
 
 
 
 
 5t> 
 
 230. 435 
 
 
 
 C~Co 
 
 43.022 
 
 M 
 
 132. 522 
 
 
 
 
 
 7j> 
 
 34.609 
 
 
 
 2+ 72?5246 
 
 2407265 
 
 M 
 
 2%. 696 
 
 
 
 4^+305. 627 
 
 281. 107 
 
 () 
 
 
 
 
 6^+186. 48 
 
 329. 70 
 
 i* 
 
 53. 263 
 
 
 
 
 
 ! 
 
 106. 526 
 
 
 
 2;+ 68783 
 
 236. 57 
 
 1* 
 
 159. 789 
 
 
 
 4^+309. 75 
 
 285.23 
 
 2e 
 
 213. 052 
 
 
 
 
 
 i 
 
 266. 315 
 
 
 
 1 Con. lot aberr. 
 
 1 From previous approi. 
 
 * From Astrand's table. 
 
 See eq. (1), page 16. 
 
24 
 
 MEMOIRS NATIONAL ACADEMY OF SCIENCES. [Voi.xiv. 
 
 (10) Hygiea. 
 PERTURBATIONS nSz, v, u, FOR 1873, SEPT. 20.4491, BER. M. T. Continued. 
 
 x-i-f+j<? 
 
 nSz [nit} 
 
 y 
 
 u 
 
 i i 
 
 x+K 
 
 log sin 
 (.x+K) 
 
 It sin (x+K) 
 
 X+*" 
 
 log cos 
 (x+K) 
 
 * cos (x+K) 
 
 x+K 
 
 log sin 
 (x+K) 
 
 * sin (x+K) 
 
 
 
 
 
 O 
 
 
 
 o 
 
 
 
 
 
 
 
 
 180.00 
 
 0. 000 n 
 
 - 40" 
 
 270.00 
 
 0.000 n 
 
 - 8" 
 
 2 
 
 
 
 
 58.608 
 
 9. 7167 
 
 +375" 
 
 296. 33 
 
 9. 952 n 
 
 -12 
 
 4 
 
 
 
 
 98. 841 
 
 9. 1867 n 
 
 - 33 
 
 238 
 
 9. 928 B 
 
 
 
 6 
 
 o 
 
 
 
 146. 28 
 
 9. 920 n 
 
 - 52 
 
 
 
 
 1 1 
 
 353. 286 
 
 9. 0679 n 
 
 - 56" 
 
 173. 425 
 
 9. 9971 
 
 - 137 
 
 80.40 
 
 9.994 
 
 + 11" 
 
 1 3 
 
 41. 102 
 
 9. 8178 
 
 + 479" 
 
 221. 824 
 
 9. 8723 n 
 
 - 193 
 
 110. 78 
 
 9.971 
 
 + 14 
 
 1 5 
 
 88.71 
 
 0.000 
 
 + 265 
 
 268. 86 
 
 8. 2988 n 
 
 - 2 
 
 156. 04 
 
 9.609 
 
 + 3 
 
 -1 1 
 
 233. 74 
 
 9. 907 n 
 
 - 85 
 
 192. 38 
 
 9. 9898 n 
 
 - 3 
 
 122. 28 
 
 9.927 
 
 + 11 
 
 -1 3 
 
 106. 53 
 
 9.982 
 
 + 41 
 
 111. 41 
 
 9. 562 n 
 
 - 13 
 
 172. 10 
 
 9.138 
 
 + 1 
 
 2 
 
 119. 27 
 
 9.941 
 
 + 18 
 
 300.02 
 
 9.699 
 
 + 3 
 
 124. 52 
 
 9.916 
 
 +103 
 
 2 2 
 
 347. 748 
 
 9. 3268 n 
 
 - 761 
 
 167. 726 
 
 9. 9900 n 
 
 -1852 
 
 79.94 
 
 9.993 
 
 + 59 
 
 2 4 
 
 35. 927 
 
 9. 7686 
 
 + 437 
 
 215. 194 
 
 9. 9123 B 
 
 - 329 
 
 104. 99 
 
 9.985 
 
 + 25 
 
 2 6 
 
 83.53 
 
 9.997 
 
 + 243 
 
 263. 148 
 
 9. 0767 n 
 
 - 15 
 
 150. 50 
 
 9.692 
 
 + 5 
 
 2 8 
 
 127.4 
 
 9.90 
 
 + 35 
 
 309. 92 
 
 9.807 
 
 + 27 
 
 
 
 
 -2 2 
 
 115. 61 
 
 9.955 
 
 + 84 
 
 88.6 
 
 8.39 
 
 
 
 0.51 
 
 7.948 
 
 + 1 
 
 -2 4 
 
 311. 82 
 
 9. 872 n 
 
 - 3 
 
 349. 03 
 
 9.992 
 
 + 6 
 
 
 
 
 3 1 
 
 
 
 
 276. 65 
 
 9.064 
 
 + 1 
 
 225. 83 
 
 9. 856 B 
 
 - 5 
 
 3 3 
 
 163. 51 
 
 9.453 
 
 + 36 
 
 341. 94 
 
 9.978 
 
 + 85 
 
 257. 49 
 
 9. 990 B 
 
 - 3 
 
 3 5 
 
 208. 94 
 
 9. 685 B 
 
 - 34 
 
 28.42 
 
 9.944 
 
 + 34 
 
 278. 56 
 
 9. 995 B 
 
 - 1 
 
 3 7 
 
 253. 08 
 
 9. 981 n 
 
 - 13 
 
 54.02 
 
 9.769 
 
 + 1 
 
 
 
 
 -3 1 
 
 
 
 
 136. 98 
 
 9. 864n 
 
 - 2 
 
 4.98 
 
 8.939 
 
 
 
 4 
 
 
 
 
 
 
 
 348.8 
 
 9. 288 n 
 
 
 
 4 2 
 
 274. 434 
 
 9. 9987 B 
 
 - 104 
 
 
 
 
 40.38 
 
 9.811 
 
 + 1 
 
 4 4 
 
 327. 82 
 
 9. 726 n 
 
 - 21 
 
 156. 79 
 
 9. 963 n 
 
 - 9 
 
 85.1 
 
 9. 998 
 
 + 1 
 
 4 6 
 
 22.1 
 
 9.58 
 
 + 5 
 
 198.96 
 
 9. 976 B 
 
 - 4 
 
 92.97 
 
 9.999 
 
 + 1 
 
 5 5 
 
 150.8 
 
 9.69 
 
 + 5 
 
 330. 79 
 
 9.941 
 
 + 10 
 
 
 
 
 5 7 
 
 196.6 
 
 9.46 B 
 
 - 2 
 
 16.85 
 
 9.981 
 
 + 8 
 
 
 
 
 
 
 
 +1648" -1079" 
 
 
 
 +550" -2684" 
 
 
 
 +236" -29" 
 
 (i O'orT 
 
 x+K' 
 
 log cos 
 (x+ff'J 
 
 k' cos (x+K') 
 
 x+K' 
 
 log sin 
 (x+K') 
 
 *' sin U+JT) 
 
 Cx+JP) 
 
 log cos 
 
 (x+K') 
 
 )c'cos(x+A") 
 
 
 
 
 
 o 
 
 
 
 
 
 
 
 
 
 
 
 270.00 
 
 0.000 n 
 
 - 6" 
 
 180.00 
 
 0.000 B 
 
 
 
 2 
 
 
 
 
 232.94 
 
 9. 902 n 
 
 g 
 
 
 
 
 4 
 
 o 
 
 
 
 281. 51 
 
 9. 991 B 
 
 - 7 
 
 
 
 
 2 
 
 292.53 
 
 9.583 
 
 + 371" 
 
 292. 573 
 
 9. 965 B 
 
 - 447 
 
 47.67 
 
 9.828 
 
 + 6" 
 
 2 2 
 
 4.7 
 
 0.00 
 
 + 2 
 
 351. 93 
 
 9. 147 B 
 
 
 
 
 
 
 2 4 
 
 41.3 
 
 9.88 
 
 + 6 
 
 41.29 
 
 9.819 
 
 + 3" 
 
 
 
 
 -2 2 
 
 123. 85 
 
 9. 746 B 
 
 - 2" 
 
 305. 02 
 
 9. 913 B 
 
 - 1 
 
 
 
 
 4 
 
 219.90 
 
 9. 885 B 
 
 - 20 
 
 
 
 
 
 
 
 4 2 
 
 104.1 
 
 9.39 B 
 
 - 1 
 
 104.23 
 
 9.986 
 
 + 4 
 
 
 
 
 4 4 
 
 154.82 
 
 9. 957 n 
 
 - 1 
 
 147 
 
 9.736 
 
 + 1 
 
 
 
 
 
 
 
 + 379" - 24" 
 
 
 
 + 8" - 469" 
 
 
 
 + 6" 
 
 (*-.)' 
 
 X+K" 
 
 log sin 
 (X+1T") 
 
 Jc" sin (x+K") 
 
 x+K" 
 
 log cos 
 <x+*") 
 
 V eta (x+K"-) 
 
 
 
 
 
 O 
 
 
 
 O 
 
 
 
 
 
 
 2 
 
 296. 23 
 
 9.953, 
 
 - 3" 
 
 112. 79 
 
 9. 588 B 
 
 - 1" 
 
 
 
 
 4 
 
 227. 15 
 
 9. 865 B 
 
 - 1 
 
 47 
 
 9.834 
 
 
 
 
 
 
 
 
 
 4" 
 
 
 
 1" 
 
 
 
 
 1 For perturbation u use factor T. 
 
No. 8.] 
 
 MINOR PLANETS LEUSCHNER, GLANCY, LEVY. 
 
 (10) Hygiea. 
 PERTURBATIONS niz, v, u, FOE 1873, SEPT. 20.4491, BER. M. T. Continued. 
 
 25 
 
 log(0-!> )rad. 9.8462 
 
 / . 
 
 
 
 
 log (> ->)* 9.692 
 
 
 
 
 
 log T 1. 3426 
 
 
 f.tj 
 
 
 
 log a sia 1" 
 
 5.183 
 
 
 
 
 
 cos t 
 
 
 1 
 
 
 
 
 
 
 
 
 n!z 
 
 
 r 
 
 u 
 
 2kaa(x.+K) + 569" Ik coe (x+^) 
 
 - 2134" 
 
 It sin (X+.BT) i + 207" 
 
 Jf cos (x+^O + 355 
 
 IV sin (x+A"0 
 
 - 461" ! JF cos (x+-K"0 6 
 
 IV sin (x+^'O 4 
 
 IV coe (x-f ") 
 
 1" 
 
 
 log IV coe (x+^O 
 log (^-^ ) If coe(x+ / K v ) 
 
 2.5502 
 2.396 
 + 249" 
 
 log IV sin (x+^0 
 log (d - >.) IV sin (x+^0 
 
 2. 664, log J'f coe (x+^"0 
 2. 510, log T. IV coe (x+-K"0 
 - 324" T. JF cos (x+^0 
 
 0.778 
 2.121 
 + 132" 
 
 logJt"sin(x+^' / ) 
 
 0.602, 
 
 r 
 
 - 2458" 
 
 u 
 
 + 339" 
 
 
 0. 294, 
 
 log * (aecs) 
 
 3.3906, 
 
 logu 
 
 2.530 
 
 (t !>,,)*. 
 
 2" 
 
 log ^ (r^) 
 
 8.0762, 
 
 
 7.713 
 
 
 
 log (l+) 
 
 9.99480 
 
 log cos a 
 
 8.798, 
 
 
 
 
 
 log cos b 
 
 9. 619, 
 
 I 
 
 + 816" 
 
 
 
 log cos c 
 
 9.958 
 
 1 Z J 1 
 
 +0. 2267 
 
 
 
 
 
 (8.3192) [n'Jz'] 
 
 +0.0058 
 
 
 
 log Ax 
 
 6. 511, 
 
 [ri^z]. 
 
 -3. 7989 
 
 
 
 log ^ J/ 
 
 7.332, 
 
 tuJz 
 
 - 3.5664 
 
 
 
 logJz 
 
 7.671 
 
 -0.00014 TKte 
 
 + 5 
 
 
 
 
 
 n*z 
 
 - 3. 566 
 
 
 Ax 
 
 -0.00032 
 
 
 
 if . ** ^' 
 
 
 Ay 
 
 -0. 00215 
 
 
 
 
 
 At 
 
 +0.00469 
 
 The computation of the geocentric place on page 26 is analogous to the usual method for 
 two body motion, the fundamental equations being (1), (2), (3). A complete set of formulae 
 and an example of the computation is also given in Memoirs of the National Academy of 
 Sciences, Vol. X, Seventh Memoir, p. 215. 
 
 CONSTANTS FOR THE EQUATOR. 
 
 
 A' yearly vr. 
 
 B' yearly var. 
 
 C' yearly vw. 
 
 log sin a log cos a 
 
 log sin b log cos b 
 
 log sin clog cos c 
 
 issao 
 
 1900.0 
 1950.0 
 
 3209833+0901399 
 321. 532+0. 01399 
 322.232+0.01399 
 
 22991g2+0901404 
 229.885-0.01405 
 230. 587+0. 01406 
 
 238657+0?01310 
 239. 312+0. 01308 
 239. 965+0. 0130S 
 
 9199914 8.799, 
 9.99914 8.797. 
 9.98915 8.795, 
 
 9.95884^9.619, 
 9.95868 9.619. 
 9.95853 9.620. 
 
 9.62355 9.958 
 9.62423 9.958 
 9.62490 9.958 
 
26 
 
 uA hod.i'i 
 
 ifj VltlHiN 
 
 MEMOIRS NATIONAL ACADEMY OF SCIENCES. 
 
 (10) Hygiea. 
 COMPARISON, OBSERVATION COMPUTATION. 1873, SEPT. 20.4491, BER. M. T. 
 
 [Vol. XIV. 
 
 
 1873 
 
 X 
 
 +3. 0709 
 
 Ber. M. T. 
 
 Sept. 20.4491 
 
 X 
 
 -1.00281 
 
 *a~f~ n j' 
 
 104?0817 
 
 dx 
 
 -0. 00032 
 
 n8z 
 
 - 3. 5660 
 
 * 
 
 +2. 0678 
 
 M=c^+n, i t+n8z 
 
 100. 5157 
 
 
 
 
 
 y 
 
 -0. 89314 
 
 dM 
 
 - 04843 
 
 Y 
 
 +0. 03260 
 
 dM' 
 
 - 29/06 
 
 jj, 
 
 -0. 00215 
 
 d<p' 
 
 + 3/15 
 
 
 -0. 86269 
 
 dv 
 dip 
 
 + 1. 8124 
 
 
 
 f X/flf^ j tff 
 
 + 68 
 
 
 
 \ fl / 
 
 1 rl 
 
 
 z 
 
 -0. 19677 
 
 ^V-SS 
 
 + 8 
 
 Z 
 
 +0. 01415 
 
 d(v-M)ldM 
 
 + 5/73 
 - 0. 0674 
 
 Jz 
 f 
 
 +0. 00469 
 -0. 17793 
 
 l hD m .dM 
 
 + 8 
 
 
 
 )2.dM' 
 
 + 1/94 
 
 log p cos S cos a 
 
 0. 31551 
 
 v-M 
 
 + 12 0/14 
 
 cos a 
 
 9. 96515 
 
 M 
 
 f+ 12 7/81 
 
 sin a 
 
 9. 58550 n 
 
 Vi~ MI 
 
 1+ 12. 1302 
 
 log p cos 8 gin a 
 
 9. 93586 n 
 
 I/-*, 
 
 112. 6459 
 
 log tg a 
 
 9. 62035 n 
 
 
 
 
 J337 21' 14" 
 
 log cos/ 
 
 9. 5S550 n 
 
 a 
 
 \ 22 h 29 m 24'. 9 
 
 log ! cos/ 
 
 8. 63170 n 
 
 Red to True a 
 
 +1.5 
 
 log (1+e, cos/) 
 
 9. 98099 
 
 True a 
 
 22 h 29 m 26 s . 4 
 
 logr 
 
 0. 51092 
 
 Obs. a (A. N. 2029) 
 
 22" 29 m 07'. 1 
 
 log (1+K) 
 
 9. 99480 
 
 
 
 logr 
 
 0. 50572 
 
 
 
 
 
 log p cos d 
 
 0. 35036 
 
 "R' 
 C" 
 
 321?1548 
 229. 5058 
 238. 9584 
 
 cos 8 
 sin 8 
 log p sin 8 
 
 9. 99864 
 8. 89852 n 
 9. 25025 n 
 
 
 >')i. ;t'>v 
 
 log tg 8 
 
 8. 89989 r 
 
 7 
 
 73. 8007 
 
 8 
 
 -4 32' 26" 
 
 '+/ 
 
 342. 1517 
 
 Red to True 3 
 
 +6" 
 
 C"+/ 
 
 351. 6043 
 
 True 8 
 
 -4 32' 20" 
 
 
 
 Obs. 8 (A. N. 2029) 
 
 -4 33' 27" 
 
 log sin a 
 
 9. 99914 
 
 
 
 log sin (A'+J) 
 
 9. 98240 
 
 -) 
 
 .!:/ 'l;i'J<-t f. 
 
 logz 
 
 0. 48726 
 
 logp 
 
 0. 35172 
 
 log sin 6 
 
 9. 95877 
 
 
 
 log sin (*'+/> 
 
 9. 48643 n 
 
 
 
 logy 
 
 9. 95092 n 
 
 (0-C) 
 
 
 
 
 Act cos 8 
 
 -19-3 
 
 log sin c 
 
 9. 62387 
 
 J8 
 
 _!' T' 
 
 logsm(C"+/) 
 
 9. 16438 n 
 
 
 
 log 2 
 
 9. 29397 n 
 
 
 
 Stlit 
 
 Given a series of observations well distributed around the orbit and extending over as long 
 an interval as is available, the elements can be corrected by the method of least squares. 
 
 For this purpose the formulae by Bauschinger 2 are convenient. The equations of condi- 
 tion are set up for the residuals in the plane of the orbit and perpendicular to the plane, as seen 
 from the earth. This resolution of the residuals is convenient because it keeps the same reso- 
 lution into components as is used in the theory of Hansen. 
 
 It is to be noticed that the elements to be used in computing the differential coefficients 
 are the finally adopted constant elements referred to the equator by the proper transformation. 
 The value of r to be used is 
 
 except in the equation 
 
 sm 
 
 sin/ 
 
 (Hansen's notation) 
 
 ' Tafel zur Berechnung der wahren Anomalie, Veroftentlichungen des Rechen-Instituts der Koniglichen Stemwarte zu Berlin No. 1. 
 
 1 tiber das Problem der Bahnverbesserung, Veroflentlichungen des Koniglichen Astronomischen Rechen-Instituts zu Berlin, No. 23, Berlin, 
 
 1903. 
 
No. 3.] 
 
 MINOR PLANETS LEUSCHNER, GLANCY, LEVY. 
 
 27 
 
 The use of ,/, r and constant elements is equivalent to the use of osculating elements for the 
 
 given date of observation. 
 
 (10) Hygiea 
 
 ^ 
 
 ^ 
 
 V 
 
 u 
 
 
 ( i 
 
 log It 
 
 K 
 
 logi 
 
 1C 
 
 log* 
 
 K 
 
 
 
 
 
 9 
 
 1.604 
 
 180.00 
 
 0.89 
 
 270.00 
 
 
 2 
 
 
 
 2.8570 
 
 254. 434 
 
 1.118 
 
 132. 16 
 
 
 4 
 
 
 
 2. 3364 
 
 130. 493 
 
 8.25 
 
 270 
 
 
 6 
 
 
 
 1.800 
 
 13.76 
 
 
 
 ndz [n8z]=2k sin (x+K) 
 
 1 1 
 
 2. 6771 
 
 37. 936 
 
 2. 1397 
 
 218. 075 
 
 1.057 
 
 125. 05 
 
 -|-( ) j_i} yjj - / k / cos (x-^-K'} 
 
 1 3 
 
 2. 8627 
 
 281. 578 
 
 2. 4135 
 
 102. 300 
 
 1.161 
 
 351. 26 
 
 +(# d<,Yk" sin (\+K") 
 
 1 5 
 
 2. 4238 
 
 165.01 
 
 1.965 
 
 345. 16 
 
 0.930 
 
 232.34 
 
 
 -1 1 
 
 2.022 
 
 24.92 
 
 0.55 
 
 343.56 
 
 1.119 
 
 273. 46 
 
 
 -1 3 
 
 1.628 
 
 93.53 
 
 1.543 
 
 98.41 
 
 0.981 
 
 159. 10 
 
 
 2 
 
 f [1.545]' 
 \ 1.320 
 
 [7. 53]' 
 12.74 
 
 0.711 
 
 193. 49 
 
 2.097 
 
 17.99 
 
 v=Ik cos (x+JiO 
 +(tf-i> )Zf sin (x+-K"') 
 
 2 2 
 
 3.5546 
 
 77.048 
 
 3. 2776 
 
 257. 026 
 
 1.777 
 
 169. 24 
 
 -(-(> i> ) J ^t" coe (x+^'O 
 
 2 4 
 
 2. 8719 
 
 321. 053 
 
 2.6054 
 
 140.320 
 
 1.412 
 
 30.12 
 
 
 2 6 
 
 2.389 
 
 204.49 
 
 2. 1033 
 
 24.100 
 
 1.034 
 
 271. 45 
 
 
 2 8 
 
 1.64 
 
 84.2 
 
 1.62 
 
 266. 70 
 
 
 
 
 -2 2 
 
 1.970 
 
 57.96 
 
 1.27 
 
 31.0 
 
 1.824 
 
 302.86 
 
 u= Ik sin (x+JQ 
 
 -2 4 
 
 0.602 
 
 90.00 
 
 0.80 
 
 127. 21 
 
 
 
 + T2V cos (x+-ST') 
 
 3 1 
 
 
 
 0.90 
 
 214. 77 
 
 0.826 
 
 163.95 
 
 
 3 3 
 
 2.100 
 
 297.46 
 
 1.95 
 
 115. 89 
 
 0.446 
 
 31.44 
 
 Where T is expressed in Julian years 
 
 
 
 
 
 
 
 
 from date of osculation. 
 
 3 5 
 
 1.841 
 
 178. 72 
 
 1.583 
 
 358.20 
 
 0.171 
 
 248.34 
 
 
 3 7 
 
 1.12 
 
 58.68 
 
 0.34 
 
 219. 62 
 
 
 
 
 -3 1 
 4 
 4 2 
 
 2. 0170 
 
 257. 208 
 
 0.42 
 
 34.68 
 
 0.673 
 0.00 
 0.270 
 
 262. 68 
 135.7 
 23.15 
 
 x=tW+y* where in s the multiples of 
 2r must be retained. 
 
 4 4 
 
 1.589 
 
 146. 42 
 
 0.97 
 
 335. 39 
 
 9.91 
 
 263.7 
 
 <> =221.811 
 
 4 6 
 
 1.14 
 
 36.5 
 
 0.66 
 
 213. 39 
 
 9.73 
 
 107. 40 
 
 
 5 5 
 
 1.038 
 
 14.0 
 
 1. 062 
 
 194.04 
 
 
 
 
 5 7 
 
 0.88 
 
 255.7 
 
 0.94 
 
 75.93 
 
 
 
 
 (-i,) or r log V 
 
 K' log it' 
 
 K' 
 
 log*' 
 
 K' 
 
 
 
 
 
 
 
 0. 799 
 
 270.00 
 
 9.690 
 
 180.00 
 
 ' 
 
 2 
 
 
 
 1.021 
 
 68.77 
 
 
 
 
 4 
 
 
 
 0.86 
 
 313. 16 
 
 
 
 
 2 
 
 2. 9862 
 
 186.00 
 
 2.6850 
 
 186. 047 
 
 0.957 
 
 301.14 
 
 
 2 2 
 
 0.18 
 
 94 
 
 0.12 
 
 81.23 
 
 
 
 
 2 4 
 
 0.88 
 
 326. 4 0. 60 
 
 326. 42 
 
 
 
 
 -2 2 
 
 0.60 
 
 66. 20 0. 11 
 
 247. 37 
 
 
 
 
 4 
 
 1.414 
 
 6.85 
 
 
 
 
 
 4 2 
 
 0.68 
 
 86. 9 0. 580 
 
 87.00 
 
 
 
 
 4 4 
 
 0.11 
 
 333. 42 0. 09 
 
 326 
 
 
 
 
 
 
 
 " v . 
 
 
 
 
 
 (>-*,) log *" 
 
 K" 
 
 logi" 
 
 K" 
 
 . 
 
 
 
 2 
 
 0.58 
 
 189. 70 
 
 0.26 
 
 6.26 
 
 
 
 
 4 
 
 9.91 
 
 14.10 
 
 9.6 
 
 194 
 
 
 
 
 COMPARISON OF THE REVISED WITH V. ZEIPEL'S ORIGINAL TABLES. 
 
 It was originally planned to conclude the example with a least squares solution of the orbit 
 on the basis of the observations used by v. Zeipel for the same purpose, and to test conclusively 
 the relative value of the revised and v. Zeipel's original tables by representing recent observa- 
 tions with both sets of elements and tables. 
 
 In the course of the computation doubt arose regarding the accuracy of some of the 
 observations selected by v. Zeipel, which led us to reject them and substitute other observa- 
 
 1 In the determination of the constant e use quantities in brackets. 
 
28 
 
 MEMOIRS NATIONAL ACADEMY OF SCIENCES. 
 
 [Vol. XIV. 
 
 tions. This substitution produced an unfavorable distribution of the observed places in the 
 orbit and the resulting least squares solution was not satisfactory. 
 
 In the meantime, pending a resumption of the least squares solution on the basis of a more 
 favorable distribution of observed places, 1 the following conclusions may be drawn regarding 
 the revised and v. Zeipel's original tables: 
 
 1. v. Zeipel's tables have been slightly improved by the correction of some numerical errors. 
 
 2. A moderate further improvement has been accomplished by an extension of the tables 
 in so far as seemed practicable without a more exhaustive and unwarranted study of the prac- 
 tical convergence of the auxiliary series, by including certain terms of higher order and degree. 
 
 With reference to the correction of the orbit and the representation of observations by a 
 least squares solution, it should be observed that 
 
 (1) A symmetrical distribution of the observed positions in the orbit is essential to coun- 
 teract the effect of neglected perturbations of higher order and degree and of major planets 
 other than Jupiter. For the Hecuba Group, in general, the mean motions of the minor planets 
 may be nearly commensurable with those of Saturn, Mars, or the Earth in the ratios 3/2, 3/1, 
 or 3/5. 
 
 (2) However accurate the initial osculating elements may be, comparatively large residuals 
 may remain on account of neglected perturbations. 
 
 Logarithmic. 
 
 TABLE A (XXXV). 
 n&z [niz] 
 
 Unlt-1" 
 
 
 Sin 
 
 te-i 
 
 W-* 
 
 -- 
 
 jf 
 
 w 
 
 to' 
 
 tf 
 
 ,. + 
 
 
 
 
 4. 1570 
 
 4. 8741 B 
 
 
 
 Jf+ 0+ 4 
 
 
 
 
 2. 7684 B 
 
 3. 3827 
 
 3. 7172 B 
 
 B* 
 
 Je+ (j-j. j 
 
 
 
 
 4. 0056 n 
 
 4.7686 
 
 
 *" 
 
 J + t>+ 4 
 
 
 
 
 4. 0766 B 
 
 4. 8295 
 
 
 
 J-+ 0+ 4 
 
 
 
 
 4. 1365 
 
 4. 8738 B 
 
 
 ' '< 
 
 + 0+24 
 
 
 
 
 3. 3345 
 
 4. 5162 B 
 
 
 
 j: + 30+24 
 
 
 
 
 4. 2240 n 
 
 4. 9611 
 
 5. 6685 B 
 
 ]J 
 
 Jtf+30+34 
 
 
 
 
 4. 0671 
 
 4. 8483 B 
 
 5. 5636 
 
 1J /3 
 
 i +50+34 
 
 
 
 
 5. 0926 n 
 
 6.0018 
 
 
 Jf 
 
 I '+50+44 
 
 
 
 
 5. 2325 
 
 6. 1714 n 
 
 
 
 i+50+54 
 
 
 
 
 4. 7675 n 
 
 5.7344 
 
 
 /" 
 
 j+50+44 2 
 
 
 
 
 3. 8050 n 
 
 4. 7998 
 
 
 
 5' 
 
 -$+ 
 
 
 
 
 3. 3112 
 
 3. 8350 H 
 
 4. 1355 
 
 
 ^r-f- t5-f- A 
 
 
 
 
 3. 2065 n 
 
 3. 7910 
 
 4. 0833 B 
 
 lj" 
 
 -}+30+ 4 
 
 
 
 
 3. 5338 
 
 4. 6236 B 
 
 
 ll' 
 
 j+30+24 
 
 
 
 
 4. 0879 
 
 5. 0382 
 
 
 1J 
 
 if+30+34 
 
 
 
 tv ' . 
 
 3. 6012 n 
 
 4. 5318 n 
 
 
 J 1 
 
 -if+30+24-J 1 
 
 
 
 
 3. 2074 
 
 4. 1925 B 
 
 
 , 
 
 , 
 
 
 9. 868 n 
 
 0. 5689 
 
 2.922 
 
 3. 4600 B 
 
 3. 3670 
 
 Jj' 
 
 + 4 
 
 
 9.482 
 
 0. 2533,, 
 
 2.673^ 
 
 3. 2959 
 
 3. 1772 B 
 
 5 1 ?' 
 
 +20+ 4 
 
 0. 746 n 
 
 1.384 
 
 3. 2927 B 
 
 4. 14906 
 
 4. 6990 B 
 
 
 
 +20+24 
 
 
 9. 788 B 
 
 2. 47560 
 
 3. 10847 n 
 
 3. 4540 
 
 3. 3960 B 
 
 n> 
 
 +20+24 
 
 0.645 
 
 1.342 B 
 
 2. 305 n 
 
 3. 6179 n 
 
 4. 4018 
 
 
 *" 
 
 t+20+24 
 
 0.326 
 
 1. 119 B 
 
 2. 935 n 
 
 3. 3017 n 
 
 4. 39206 
 
 
 
 +20+24 
 
 
 
 3. 4276 B 
 
 4. 23764 
 
 4. 76933n 
 
 
 '!')' 
 
 +20+34 
 
 0. 28 B 
 
 1.102 
 
 3. 1738 
 
 3. 5449 n 
 
 3. 8446 n 
 
 
 rjij' 3 
 
 +40+24 
 
 
 
 3.6004 
 
 4. 27485 
 
 
 
 1 
 
 (.(-40+34 
 
 9.057 
 
 0. 692 B 
 
 3. 10161 
 
 3. 9302 B 
 
 4. 52415 
 
 4. 78162 B 
 
 ,V 
 
 +40+34 
 
 
 
 4. 0519 n 
 
 3. 7975 
 
 
 
 yf 
 
 +40+34 
 
 
 
 4. 1385 n 
 
 4. 6961 
 
 
 
 
 +40+34 
 
 
 
 4. 2431 B 
 
 5.1290 
 
 
 
 17 
 
 +40+44 
 
 9.500 B 
 
 0.522 
 
 2. 9351 B 
 
 3. 8035 
 
 4. 41616 n 
 
 4. 63017 
 
 if 
 
 +40+44 
 
 
 
 3. 7714 
 
 4. 2108 B 
 
 
 
 Iff'* 
 
 +40+44 
 
 
 
 4. 4165 
 
 5. 0931 B 
 
 
 
 Pi) 
 
 j-j-4,j-|-44 
 
 
 
 4. 1524 
 
 5. 0661 B 
 
 
 
 ,y 
 
 +40+54 
 
 
 
 4. 0588 B 
 
 4. 8136 
 
 
 
 ' Since 1913, when the revision of the tables was concluded, Miss Glancy has continued the problem of ( 10) Hygica independently at the Observa 
 irio National, Cdrdoba, with the following highly satisfactory results, which substantiate further the increased accuracy of the revised table: 
 
 va- 
 
 torio Nacional, Cdrdoba, with the following highly satisfactory results, which substantiate further the increased accuracy of the revised tables 
 (1) The original osculating elements and the revised tables resulted in a greatly improved representation of the selected observations (1849-188i) 
 over the representation obtained with the original tables. (2) After the correction of the original osculating elements by least squares solution 
 (a) on the oasis of v. Zeipel's tables and residuals, (6) on the basis of the residuals resulting from the revised tables, the representation ol the 
 selected observations was equally satisfactory; but 3 later observations, taken in 1910, 1914, and 1917, are represented far better by the revised 
 tables and corresponding elements than by the original tables and corresponding elements, (of. Astronomical Journal, Vol. 32, p. 27, No. 748, 
 January 1919) A. O. Leuschner. 
 
Ko.3.1 MINOR PLANETS LEUSCHNER, CLANCY, LEVY. 
 
 Logarithmic. TABLE A (XXXV) Continued. 
 
 29 
 
 Unit-l" 
 
 
 Sin - K"- 1 M 
 
 
 V 1C* 
 
 J 1J 
 
 f-J-4^-f-3J 2 
 
 
 
 3.2322, 
 
 4.2342 
 
 
 
 
 c-f~4iJ-|-4J E 
 
 
 
 2.744, 
 
 3. 0962 
 
 
 
 ,/S 
 
 t+6t>+4J 0.28 0.64, 
 
 3.8027 
 
 4. 77998, 5. 52852 
 
 
 - -/ 
 
 -j-6iJ-(-5J 
 
 0.596, 
 
 1.070 
 
 3. 9374, 
 
 4. 94342 5. 70347, 
 
 
 ij l 
 
 c -\-M+<}J 
 
 0.255 
 
 0.8, 
 
 3.4684 
 
 4.50125, 
 
 5.27451 
 
 
 
 +6t?+5J-J 
 
 8.8 
 
 9-3, 
 
 2.415 
 
 3.4823, 
 
 4.2931 
 
 
 V* 
 
 e+8^+5J 
 
 
 
 4.5564 
 
 5. 4999, 
 
 
 
 "7 
 
 -j_g^_j_6J 
 
 
 
 4.8668, 
 
 5.8416 
 
 
 
 
 _)-g^_(-7J 
 
 
 
 4.6990 
 
 5. 7030, 
 
 
 
 i* 
 
 e-(-8i>+8J 
 
 
 
 4.0631, 
 
 5.0844 
 
 
 
 j ^' 
 
 j-Lg^-LgJ^ 1 
 
 
 
 3.5829 
 
 4. 6352, 
 
 
 
 ft +8e>+7J-J 
 
 
 
 3.3768, 
 
 4.4540 
 
 
 
 l" 
 
 - +2<J 
 
 0.606 
 
 1.422, 
 
 3. 2132 
 
 3. 6657, 
 
 3.9260 
 
 
 r ' 
 
 + 2l?+ J 
 
 0. 791, 1. 690 
 
 3. 3777 B 
 
 3.8866 
 
 4. 72168 
 
 
 1 
 
 - +2tf+2J 
 
 0. 418 1. 365, 
 
 2.894 
 
 3. 4616.., 
 
 3.8078 
 
 
 
 - +20+ A-I 
 
 9.34 
 
 0.28 B 
 
 2.938 
 
 3. 4714, 
 
 3. 7862 
 
 
 /i 
 
 - +4^+ A 
 
 ' t '-> 
 
 ' -* ,' 
 
 3.5208 
 
 4.07255 
 
 
 
 7 r/' 3 
 
 
 
 
 3.4965, 
 
 4.59582 
 
 
 
 r*r' 
 
 _ -j-4ty+3J 
 
 
 
 3.2416 
 
 4. 5467, 
 
 
 
 1)' - +4tf+4J 
 
 
 
 2.430, 
 
 3.9848 
 
 
 
 ,' - e+4d+2J-2 
 
 
 
 3.5496 
 
 4.19852, 
 
 
 
 
 - f+4<5+3J-J 
 
 
 
 3. 3247, 
 
 4.05994 
 
 
 
 55' 
 
 ^ +3l j + 2J 
 
 
 
 
 3. 6731 
 
 4.0029, 
 
 
 
 |+3tJ-(-3J 
 
 
 
 
 2.3528 
 
 3. 2475, 
 
 3.9005 
 
 ij' 
 
 ^+3i>+3J 
 
 
 
 
 3. 6181, 
 
 4.2122 
 
 
 f 
 
 -j-3,y-j-3J 
 
 
 
 
 3. 4072, 
 
 4.4000 
 
 
 
 i + 3l?+4J 
 
 
 
 
 3.5244 
 
 4.4012, 
 
 
 7/ ' 
 
 ^+5t>+4J 
 
 
 
 
 3.3533 
 
 4. 4231, 
 
 5.2725 
 
 ^ 
 
 i+5t>+5J 
 
 
 
 
 3. 1780, 
 
 4.2730 
 
 5. 1359, 
 
 q'l 
 
 ,-)-7i)4-5J 
 
 
 
 
 4.2775 
 
 5.4708, 
 
 
 jj -' 
 
 i + 7l? + 6J 
 
 
 
 
 4. 4051, 
 
 5. 6177 
 
 
 >z 2 
 
 i*-J-7iJ+7J 
 
 
 
 
 3.92% 
 
 5.1605, 
 
 
 ij 
 
 2^+2^+2^ 
 
 
 9.486 
 
 2. 1744, 2. 708 
 
 2. 889, 2. 599, 
 
 ' / 
 
 2-i-2i'+3J 
 
 
 1.946, 
 
 2. 501 2. 516, 
 
 ? ?' 
 
 2 -f 4^-j-3J 
 
 8. 8, 0. 561 
 
 2. 789, 
 
 3.5813 
 
 4. 1074, 
 
 
 2j-j-4<>-[-4j 
 
 
 8.90, 
 
 9.599 
 
 1.711 
 
 2.5795, 
 
 3. 1726 
 
 * 
 
 2 +4^+4J 
 
 9.2 
 
 0.34, 
 
 2.618 
 
 3.4962, 
 
 4.0890 
 
 
 ^' 
 
 2-j-6J+5J 
 
 
 9. 819, 
 
 0.5840 
 
 2. 7821 
 
 3. 7794, 
 
 4.51865 
 
 1 
 
 2 e +6<J+6J 
 
 
 9.653 
 
 0.4645, 
 
 2. 5979, 
 
 3.6265 
 
 4.38424, 
 
 
 Sr-j-otf + oJ 
 
 
 
 
 1.2340 
 
 2. 1166, 
 
 2.7076 
 
 T' 
 
 i-|-7i)4-6J 
 
 
 
 
 2. 3679 
 
 3.3518, 
 
 4.0587 
 
 1 
 
 |+7!> + 7J 
 
 
 
 
 2.1758, 
 
 3.1926 
 
 3.9204, 
 
 
 (t) #) coe 
 
 
 
 
 
 
 
 i; 
 
 
 
 0. 1021, 
 
 0.728 
 
 2.8978, 
 
 3.4504 
 
 3.7168, 
 
 
 
 
 
 1.377, 
 
 2.346 
 
 3. 8211, 
 
 4.6762 
 
 
 'i" 
 
 f 
 
 1.941, 
 
 2.815 
 
 4. 4076, 
 
 5. 1971 
 
 
 
 
 
 
 1.364 
 
 2.220, 
 
 4.4076 5.1971, 
 
 5. 7086 
 
 
 ^' 
 
 e-f J 
 
 9.658 
 
 0. 774, 
 
 2. 7836 3. 3840, 
 
 3.6946 
 
 
 '/' T / 
 
 + -1 
 
 1.863 
 
 2.755, 
 
 4. 2546 5. 0814, 
 
 
 
 * 
 
 *4" J 
 
 L844 
 
 2.642, 
 
 4.1953 
 
 4. 9770, 
 
 
 
 f *,' 
 
 + J 
 
 1. 170 n 
 
 2.049 
 
 4. 3715, 
 
 5. 1770 
 
 5.6975, 
 
 
 v 
 
 + 2J 
 
 1. 742 B 
 
 2.574 
 
 4.0203, 
 
 4.8466 
 
 
 
 j 3 V 
 
 t+ J 0. 716 
 
 1.65, 
 
 4. 0809 4. 8829, 
 
 5.4008 
 
 
 j*, 
 
 + J+^ 
 
 1.00, 
 
 1.89 
 
 4. 3427, 5. 0837 
 
 5.5553, 
 
 
 v 
 
 - J- J 
 
 1.562 
 
 2.455, 
 
 3. 9535 4. 7803, 
 
 
 
 ,1 
 
 2: 
 
 9.801 
 
 0.43, 
 
 2. 5842 3. 1493, 
 
 3.4158 
 
 
 1 1 
 
 2+ J 
 
 9. 357 n 
 
 0.473 
 
 2.4548, 
 
 3.0830 
 
 3. 3936, 
 
 
 \v W 0' *-** 
 
 7 * 
 
 
 9.56, 
 
 0.42 
 
 
 
 
 + J 
 
 
 9.43 
 
 0.32, 
 
 
 
 
 where C,, 
 
 sin Arg.4-( I )-iJ )J'r / P^ / 9; 2 Cj coe 
 2 , C 3 represent the respective coefficients. 
 
 sin Arg. 
 
30 
 
 MEMOIRS NATIONAL ACADEMY OF SCIENCES. 
 
 IVol. XIV. 
 
 Logarithmic. 
 
 TABLK B (XXXVIII). 
 *() 
 
 Unit 1 radian. 
 
 
 Cos 
 
 ,, 
 
 JM 
 
 -. 
 
 M 
 
 '** 
 
 *-. 
 
 * 
 
 w 
 
 te* 
 
 
 
 
 
 1.5 
 
 3. 909 B 
 
 4.960 
 
 6. 6748 B 
 
 7. 2764 
 
 7. 540 B 
 
 7.31 
 
 ** 
 
 
 
 2.0 
 
 4. 644 B 
 
 5.160 
 
 6. 150 
 
 8. 048 B 
 
 8.838 
 
 8. 655 n 
 
 8. 100 n 
 
 V* 
 
 
 
 1.9 
 
 3.41 n 
 
 4.75 n 
 
 6.509 
 
 8. 2077 B 
 
 8.994 
 
 8. 919 B 
 
 
 P 
 
 
 
 
 2.83 n 
 
 5.146 
 
 6. 299 B 
 
 7. 994 
 
 8. 740 n 
 
 8.656 
 
 
 
 * 
 
 
 2.34 B 
 
 4.446 
 
 4.57 
 
 6. 728 B 
 
 8. 4022 
 
 9. 1999 B 
 
 9.0854 
 
 8.079 
 
 Tj Tj 
 
 2i> 
 
 1.6 
 
 2.6 B 
 
 5.744 
 
 6. 535,, 
 
 8.3811 
 
 9. 1031 B 
 
 9. 0128 
 
 
 
 if 
 
 2<>+ A 
 
 
 0.8 n 
 
 3.068 
 
 5. 2988 
 
 7. 2212 B 
 
 7. 3772 
 
 8. 0372 
 
 8. 764 B 
 
 8.668 
 
 Tj 1) 
 
 2<?+ J 
 
 2.32 B 
 
 3.30 
 
 5. 886 B 
 
 6.718 
 
 8. 5059 B 
 
 9.2804 
 
 9. 201 7 B 
 
 
 
 1)'* 
 
 2t?+ 4 
 
 
 
 5. 301 B 
 
 6.149 
 
 8. 2302 B 
 
 9. 0154 
 
 8. 938 B 
 
 
 
 P y' 
 
 20 + A 
 
 
 
 
 
 8. 5592 
 
 9. 3245 B 
 
 9. 2428 
 
 
 
 y 3 
 
 20 +2J 
 
 2.48 
 
 3.40 n 
 
 5.422 
 
 6. 292 B 
 
 7.476 
 
 8. 664 n 
 
 8.636 
 
 
 
 Tj 
 
 20+2J 
 
 
 1.22 
 
 2.94 n 
 
 5. 1206 B 
 
 7.6416 
 
 7. 9638 n 
 
 7.083 B 
 
 8.645 
 
 8. 582 n 
 
 1) I)' 2 
 
 2#+24 
 
 1.9 
 
 3.0 n 
 
 5.442 
 
 6. 328 B 
 
 8. 0915 B 
 
 8. 630 B 
 
 8.742 
 
 
 
 3 V 
 
 20+2J 
 
 
 
 
 
 8. 5904 
 
 9. 3489 
 
 9. 8024 n 
 
 9.6532 
 
 
 TJ ij 
 
 2t>+3J 
 
 2.04 B 
 
 3.00 
 
 4.98 B 
 
 5.89 
 
 8. 0326 ! 8. 1973 n 
 
 7.69 
 
 
 
 fi) 
 
 2i>+ J-J 
 
 
 
 4.51 
 
 5.42 B 
 
 8. 1011 
 
 8. 873 B 
 
 8.792 
 
 
 
 .'2 _/ 
 y v 
 
 20+2J-.T 
 
 
 
 4.04 B 
 
 5.00 
 
 6.89 B 
 
 8.182 
 
 8. 158 B 
 
 
 
 ,' 
 
 40+24 
 
 
 2.66 n 
 
 2.7 
 
 6. 1031 
 
 8. 4188 B 
 
 8. 5297 
 
 6.0 
 
 7.90 n 
 
 
 Tj Tj 
 
 4<?+3J 
 
 
 2.72 
 
 4.369 
 
 6. 2526 B 
 
 8. 5594 
 
 8. 7988 B 
 
 7.94 B 
 
 8.287 
 
 8.210 
 
 !J 2 
 
 40+44 
 
 
 2.20 B 
 
 4. 624 B 
 
 5.824 
 
 8. 0924 B 
 
 8. 4333 
 
 7.24 
 
 7.74 B 
 
 8. 044 B 
 
 P 
 
 4<H-3J-.y 
 
 
 1.5 B 
 
 2.45 
 
 4.68 
 
 7. 1747 B 
 
 7.301 
 
 8.111 
 
 8. 127 B 
 
 
 
 
 6.J+34 
 
 
 
 5. 301 B 
 
 6.149 
 
 9. 1294 B 
 
 9. 7728 
 
 9. 6609 n 
 
 
 
 
 6i>4~4J 
 
 
 
 5.92 
 
 6.74 B 
 
 9. 4432 
 
 0. 14644 B 
 
 0. 05077 
 
 
 
 7)V 
 
 6i5+5J 
 
 2.0 B 
 
 3.0 
 
 5.93 n 
 
 6.79 
 
 9. 2774 B 
 
 0. 03298 
 
 9. 9494 n 
 
 
 
 n* 
 
 6<>+6J 
 
 2.0 
 
 3.0 B 
 
 5.420 
 
 6. 292 n 
 
 8.634 
 
 9. 4351 n 
 
 9. 3608 
 
 
 
 ;? i 7 
 
 6<>-|-4J 2" 
 
 
 
 4.04 B 
 
 5.00 
 
 8. 272 B 
 
 9. 1028 
 
 9. 0334 n 
 
 
 
 J >! 
 
 e^+SJ-J 
 
 
 
 4.51 
 
 5.42 B 
 
 8. 0554 
 
 8. 926 B 
 
 8.864 
 
 
 
 
 (i>-<5 ) sin 
 
 
 
 
 
 
 
 
 
 
 ^' 
 
 J 
 
 
 
 2.60 n 
 
 4.71 
 
 5.94 B 
 
 6.507 B 
 
 6.606 
 
 
 
 ,/ 
 
 2.+ 4 
 
 
 1.36 
 
 2.48 
 
 4.49 
 
 5. 255 n 
 
 5.51 
 
 5. 25 B 
 
 
 
 * 
 
 2t>+2J 
 
 
 1.82 B 
 
 2.42 
 
 4.64 B 
 
 5.350 
 
 5.51 B 
 
 5.16 
 
 
 
 ,'2 
 
 40+24 
 
 
 2.34 
 
 3.00 
 
 5.392 
 
 6. 179 B 
 
 6. 528 n 
 
 6.665 
 
 
 
 Tj if 
 
 4i?-|-3^i 
 
 
 2.89 B 
 
 3.46 
 
 5. 702 B 
 
 6.467 
 
 6.851 
 
 6. 979 B 
 
 
 
 I* 
 
 4i?-j-4^ 
 
 
 2.66 
 
 3. 459 B 
 
 5.357 
 
 6. 127 B 
 
 6. 530 B 
 
 6.653 
 
 
 
 1* 
 
 
 
 
 2.08 B 
 
 2.08 
 
 
 5. 546 B 
 
 5.546 
 
 
 
 if* 
 
 
 
 
 2.54 
 
 2.54 n 
 
 
 5. 396 B 
 
 5.396 
 
 
 
 !l' 
 
 4 
 
 
 
 2.5 n 
 
 2.5 
 
 
 5.776 
 
 5. 776 n 
 
 
 
 
 
 m' 3 
 
 m' 3 
 
 m' 3 , m' 2 
 
 m' 3 , m' 3 
 
 m' 2 , m' 
 
 m "' m/ 
 
 m' 2 , m' 
 
 m /1 , m' 
 
 m' 1 , m 
 
 1 cos Arg.+( t ?-tf )Jit'*jP., / 9./.C 2 sin Arg.-f (^-t 
 where C,, C 2 , C 3 represent the respective coefficients. 
 
 cos Arg. 
 
No. 3.] 
 
 MINOR PLANETS LEUSCHNER, GLANCY, LEVY. 
 
 TABLE C (XLIII). 
 
 31 
 
 Ixxrarithmfc. 
 
 Unit-l" 
 
 
 Cos 
 
 - 
 
 2 
 
 - 
 
 w* 
 
 tr 
 
 
 
 
 
 
 8.72 
 
 9.88, 
 
 1.6349 
 
 2. 1070, 
 
 2.2333 
 
 f 
 
 
 9.80 
 
 0.212, 
 
 
 2.759 
 
 3.4922, 
 
 
 V* 
 
 
 8.9 
 
 9.23 
 
 
 2.937 
 
 3.6295, 
 
 
 ' 
 
 J 
 
 9.66n 
 
 9.78 
 
 
 2.937, 
 3.1136, 
 
 3.6295 
 3.8440 
 
 
 If" 
 
 M 
 
 0.556, 
 
 1.204 
 
 3. 2111, 
 
 3.7970 
 
 
 
 If 
 
 2tf+ 4 
 
 
 0.504, 
 
 2.3472 
 
 2.456 n 
 
 2.686, 
 
 3.4735 
 
 rY 
 
 2i+ ^ 
 
 0.997 
 
 1.711, 
 
 3.6559 
 
 4. 3103, 
 
 
 
 7" 
 
 2iJ+ 4 
 
 0.438 
 
 1.220, 
 
 3.3654 
 
 4. 0763, 
 
 
 
 i* * 
 
 2<>+ A 
 
 
 
 3.6975, 
 
 4. 3810 
 
 
 
 
 i) 
 
 20 +2 J 
 
 
 0.438 
 
 2.952, 
 
 3. 2529 
 
 3.0689, 
 
 3.3979, 
 
 f 
 
 2.5 +2 J 
 
 0.732, 
 
 1.497 
 
 3. 2410, 
 
 4.0643 
 
 
 
 7-j" 
 
 2tf+2J 
 
 0.772, 
 
 L589 
 
 3. 4136 
 
 4.0723 
 
 
 
 A 
 
 2J+2J 
 
 
 
 3.9048 
 
 4.5649, 
 
 4.9303 
 
 
 V 
 
 2i>+34 
 
 0.505 
 
 1.344, 
 
 3. 4757, 
 
 2.783 
 
 
 
 /*5 
 
 2+ J-2 1 
 
 9.33, 
 
 0.15 
 
 2.938, 
 
 3.5830 
 
 
 
 ? V 
 
 2^+24 -J 
 
 9.20 
 
 0.10, 
 
 2.0251 
 
 3.2961, 
 
 
 
 ?" 
 
 4J+24 
 
 8.9 
 
 1. 2819, 
 
 3.5514 
 
 3. 6173, 
 
 3. 8147 
 
 
 it* 
 
 w+w 
 
 9.75, 
 
 1.5024 
 
 3.7885, 
 
 4.1394 
 
 4. 3110, 
 
 
 r,> 
 
 4i>+4J 
 
 9.98 
 
 1.1342, 
 
 3.4007 
 
 3.9091, 
 
 4.1480 
 
 
 'V 
 
 4j+3^-J 
 6^+SJ 
 
 0.438 
 
 9.64, 
 1.220, 
 
 2.305 
 4.2675 
 
 2.542, 
 4.7993, 
 
 2. 749, 
 
 
 u? 
 
 W+4J 
 
 L125, 
 
 1.862 
 
 4. 6479, 
 
 5.2324 
 
 
 
 *v 
 
 6i)+5J 
 
 1.198 
 
 1.947, 
 
 4.5397 
 
 5.1768, 
 
 
 
 7* 
 
 6J-I-6J 
 
 0.732, 
 
 1.508 
 
 3.9457, 
 
 4.6328 
 
 
 
 F *' 
 
 6rf+44-^ 
 
 9.20 
 
 0.10, 
 
 3.4099 
 
 4. 1710, 
 
 
 
 w 
 
 6tf+5J-J 
 
 9.70, 
 
 0.56 
 
 3.2601, 
 
 4.0542 
 
 
 
 
 
 
 
 
 
 
 
 9 v' 
 
 i+ i> 
 
 
 
 
 3.4878, 
 
 4.1106 
 
 
 
 i+ + J 
 
 
 
 8.3. 
 
 2.2106 
 
 2. 7179, 
 
 2.919 
 
 f 
 
 i*+ J+ 4 
 
 
 
 
 3. 5709, 
 
 4.2261 
 
 
 , 
 
 1 + tf+ J 
 
 
 
 
 3.4507 
 
 4.1296, 
 
 
 V 
 
 <H- rf+ ^ 
 
 
 
 
 3.5100 
 
 4. 1837, 
 
 
 ?V 
 
 v+ <J+2J 
 
 
 
 
 2. 579, 
 
 3.9270 
 
 
 7' 
 
 < t+3*+24 
 
 
 
 0.08 
 
 3.6873 
 
 4. 1471, 
 
 4.7839 
 
 q 
 
 . E+3J+3J 
 
 
 
 9.5 
 
 3. 5727, 
 
 4.1511 
 
 4.7545, 
 
 5" 
 
 < t+5tJ+3J 
 
 
 
 
 4.5568 
 
 5. 1414, 
 
 
 -M' 
 
 - t+5t>+4J 
 
 
 
 
 4. 7261, 
 
 5.4067 
 
 
 i 3 
 
 1 f+StJ+54 
 
 
 
 
 4.2862 
 
 5.0418, 
 
 
 J 3 
 
 1 t+5i+4J-J 
 
 
 
 
 3.2570 
 
 4.0005, 
 
 
 1 
 
 -i+ * 
 
 
 
 L086, 
 
 2.7090 
 
 3. 3467, 
 
 3.7098 
 
 1 
 
 -i+ l>+ J 
 
 
 
 0.88 
 
 2. 1967, 
 
 3.0952 
 
 3.5836, 
 
 I* 
 
 -i-t+3*+ J 
 
 
 
 
 2.514 
 
 4,1049, 
 
 
 ijr 
 
 - t+3J+2J 
 
 
 
 
 4.0853 
 
 3.9122 
 
 
 f 
 
 -i+3J+3J 
 
 
 
 
 3.8341, 
 
 3. 8118 
 
 
 f 
 
 -|+3J-f-2^-J 
 
 
 
 
 2.416 
 
 3.6926, 
 
 
 >! 
 
 i 
 
 
 9.62 
 
 0.58, 
 
 2.143, 
 
 2.682 
 
 2.9151, 
 
 r 
 
 e+ J 
 
 
 9.04, 
 
 9.9 
 
 2.061 
 
 2.666, 
 
 2.9477 
 
 if' 
 
 +2l>+ J 
 
 0.444 
 
 1.1661* 
 
 3.0588 
 
 3.8035, 
 
 4.2554 
 
 
 
 t+2^+2^ 
 
 
 9.487 
 
 2.1744, 
 
 2.7280 
 
 2.972, 
 
 2.976 
 
 i 3 
 
 +2^^-24 
 
 0.344, 
 
 L1143 
 
 2.692, 
 
 3.5334 
 
 4.0772, 
 
 
 *" 
 
 +2J+2J 
 
 0.025, 
 
 0.828 
 
 2.634 
 
 3.0726 
 
 4.0416, 
 
 
 j 3 
 
 +2^+24 
 
 
 
 3.1265 
 
 3.8806, 
 
 4.3473 
 
 
 IT* 
 
 +2<>+3J 
 
 9.98 
 
 0.811, 
 
 2.873, 
 
 3. 1697 
 
 3.5856 
 
 
 t7 
 
 E+4J+24 
 
 1.105 
 
 L89, 
 
 2.864 
 
 4. 3477, 
 
 
 
 < 
 
 +4t>+34 
 
 8.8, 
 
 0.398 
 
 2.8000, 
 
 3.5327 
 
 4.0065, 
 
 4.3207 
 
 iV 
 
 +4t>+3J 
 
 1.260, 
 
 2.083 
 
 3.0931 
 
 4.4160 
 
 
 
 i" 
 
 +4t>+34 
 
 
 
 3.8375 
 
 4.0446, 
 
 
 
 >* i 7 
 
 +4tf+3J 
 
 0.267 
 
 1.15, 
 
 3.9421 
 
 4. 6972, 
 
 
 
 i 
 
 +4t?+4J 
 
 9.19 
 
 0.248, 
 
 2.6356 
 
 3. 4317, 
 
 3.9469 
 
 4.2558, 
 
 f 
 
 +4tf+4J 
 
 0.774 
 
 L66, 
 
 3.0934, 
 
 3.7866, 
 
 
 
 i" 
 
 +4J+4J 
 
 
 
 4.1154, 
 
 4.5547 
 
 
 
 j 3 '? 
 
 +4^+4^ 
 
 0.455, 
 
 1.32 
 
 3.8518, 
 
 4.6436 
 
 
 
 v 
 
 +4^+54 
 
 
 
 3. 7579 
 
 4.3244, 
 
 
 
 /N 
 
 +4tf+3J-J 
 
 
 
 3.0030 
 
 3.8869, 
 
 
 
32 
 
 MEMOIRS NATIONAL ACADEMY OF SCIENCES. 
 
 TABLE C (XLIII) Continued. 
 
 Logarithmic. 
 
 [Vol. XIV. 
 
 Unlt-1" 
 
 
 Cos 
 
 w-> 
 
 w-> 
 
 w-i 
 
 to' 
 
 w 
 
 w' 
 
 f if 
 
 s+40+44-1 
 
 
 
 2. 4425 
 
 1. 85 n 
 
 
 
 ^ 
 
 +C^+4J 
 
 9.98 n 
 
 0.480 
 
 3. 5016,, 
 
 4. 3723 
 
 4. 9952 n 
 
 
 w' 
 
 e+6<)+5J 
 
 0.296 
 
 0. 823 B 
 
 3. 6369 
 
 4. 5582 n 
 
 5. 2093 
 
 
 f 
 
 +G^+6J 
 
 9.95 n 
 
 0.538 
 
 3. 1685 n 
 
 4. 1334 
 
 4. 8131 n 
 
 
 f 
 
 f+Gtf+SJ-I 1 
 
 8.5 n 
 
 9.15 
 
 2. 114 n 
 
 3. 0881 
 
 3. 7886 n 
 
 
 ^ 
 
 +8tf+5J 
 
 
 
 4. 2554 n 
 
 4. 9349 
 
 
 
 ,," 
 
 +8!?+6J 
 
 1.320 
 
 2. 152 n 
 
 4. 5657 
 
 5. 3010 n 
 
 
 
 ,V 
 
 e+8,?+7J 
 
 1. 228 B 
 
 2.093 
 
 4. 3995 n 
 
 5. 1827 
 
 
 
 1* 
 
 e+8,9+84 
 
 0.648 
 
 1.54 n 
 
 3.7543 
 
 4. 5812 n 
 
 
 
 P l' 
 
 +80+64-.? 
 
 
 
 3. 2818 n 
 
 4. 1442 
 
 
 
 A 
 
 e+Stf+74-2 1 
 
 
 
 3. 0763 
 
 3. 9759 n 
 
 
 
 *" 
 
 - +2t5 
 
 0.305 
 
 1. 1007 
 
 2.912 
 
 3. 4958 n 
 
 3. 8151 
 
 
 11' 
 
 - +2<J+ 4 
 
 0. 490 n 
 
 1. 3330 
 
 3. 0166 n 
 
 3. 7273 
 
 4. 3119 
 
 
 V* 
 
 - e+2t?+24 
 
 0.117 
 
 0. 982 n 
 
 2. 288 n 
 
 3. 2375 B 
 
 3. 7892 
 
 
 f 
 
 - +20+ J-2 
 
 9.04 
 
 9.96 n 
 
 2.636 
 
 3. 2817 n 
 
 3. 6568 
 
 
 ," 
 
 - +40+ J 
 
 
 
 3. 2197 
 
 3. 9650 n 
 
 
 
 ,," 
 
 - +40+24 
 
 1. 146 n 
 
 1.89 
 
 3. 0204 
 
 4. 2441 
 
 
 
 *V 
 
 - +40+34 
 
 1.005 
 
 1.78 n 
 
 3. 5247 n 
 
 4. 0012 n 
 
 r f \i 
 
 
 1* 
 
 - +40+44 
 
 0.290 n 
 
 1.15 
 
 3. 1793 
 
 2.982 
 
 n- ^ 
 
 
 f rf 
 
 - +40+24 -.T 
 
 
 
 3. 2486 
 
 4. 0585 n 
 
 :.-W- 
 
 
 S, 
 
 - +40+34-1 
 
 9.98 n 
 
 0.8 
 
 2. 957 n 
 
 3. 8580 
 
 
 
 y 
 
 |+ 0+ 4 
 
 
 
 9.0 
 
 2. 3363 
 
 3. 0704 B 
 
 3. 5111 
 
 r,' 
 
 i*+ 0+24 
 
 
 
 9.5 
 
 1.500 
 
 2. 3585 
 
 3. 1842 B 
 
 1)1)' 
 
 -|+30+24 
 
 
 
 
 2.779 
 
 3. 7820 n 
 
 
 
 ^+30+34 
 
 
 
 9.28 
 
 2. 1614^ 
 
 3. 0257 
 
 3. 6491 B 
 
 ? 
 
 | +30+34 
 
 
 
 
 1.32 
 
 2.966 
 
 
 l" 
 
 |+30+34 
 
 
 
 
 3.3450 
 
 4. llll n 
 
 
 ? 
 
 |+30+34 
 
 
 
 
 3. 2309 
 
 4. 1965 n 
 
 
 ?Y 
 
 |+30+44 
 
 
 , 
 
 
 3. 2994 n 
 
 4. 1520 
 
 
 T,' 
 
 |+50+44 
 
 
 
 1.017 
 
 3. 1617 n 
 
 4. 1967 
 
 5. 01GO n 
 
 T) 
 
 |+50+54 
 
 
 
 0.88 n 
 
 2. 9688 
 
 4. 0380 n 
 
 4. 8781 
 
 *" 
 
 ^+70+54 
 
 
 
 
 4. 0855 B 
 
 5. 2422 
 
 
 >y 
 
 |+70+64 
 
 
 
 
 4. 1991 
 
 5. 3823 n 
 
 
 ^ 
 
 fs+70+74 
 
 
 
 
 3. 7114 n 
 
 4. 9188 
 
 
 ; 3 
 
 |+7d+64-J 
 
 
 
 
 2. 615 n 
 
 3. 8317 
 
 
 riV 
 
 -!e+ tf 
 
 
 
 
 3. 2411 
 
 3. 7872 n 
 
 
 tf 
 
 -i+ 0+4 
 
 
 
 
 2. 819 n 
 
 3. 4476 
 
 
 ? 
 
 -^+ 0- 1 
 
 
 
 
 2. 9181 
 
 3. 4813 
 
 
 >? 
 
 2 
 
 B 98H.( 
 
 
 
 2. 364 n 
 
 3. 0737 
 
 
 v 
 
 2s+ 4 
 
 XX .0 
 
 
 
 2.624 
 
 3. 3489 B 
 
 
 ," 
 
 2s+ 24 
 
 
 
 
 2. 207 n 
 
 2.978 
 
 
 f 
 
 2+ 4+1 
 
 
 
 
 2. 620 n 
 
 3. 2765 
 
 
 1? 
 
 2s+20+24 
 
 
 
 9.8 n 
 
 1.63 
 
 2. 362 n 
 
 2. 873 
 
 *' 
 
 2 +20+34 
 
 
 
 9.5 
 
 1.796 
 
 2. 303 n 
 
 2. 1007 
 
 -M' 
 
 2 +40+34 
 
 
 
 
 1.93. 
 
 2.700 
 
 
 
 2 +40+44 
 
 
 8.7 
 
 8.8 
 
 1. 5802 n 
 
 2. 4158 
 
 2. 9867 B 
 
 * 
 
 2+40+44 
 
 
 
 
 2.330 
 
 3. 1764 n 
 
 
 ," 
 
 2t+40+44 
 
 
 
 
 3. 1079 
 
 3. 9008 n 
 
 
 ;' 
 
 2e+40+44 
 
 
 
 
 2.736 
 
 3. 6809 n 
 
 
 V 
 
 2 4 +40+54 
 
 
 
 
 2. 9881 n 
 
 3.8425 
 
 
 V 
 
 2+60+54 
 
 
 9.64 
 
 0.53 
 
 2. 652 n 
 
 3.6204 
 
 4. 3279 B 
 
 >j 
 
 2+60+64 
 
 
 9.48 n 
 
 0.36 n 
 
 2. 4419 
 
 3. 4512 B 
 
 4. 1892 
 
 >)" 
 
 2+80+64 
 
 
 
 
 3. 6135 n 
 
 4. 6784 
 
 
 -M' 
 
 2 +80+74 
 
 
 
 
 3. 7124 
 
 4. 8075 n 
 
 
 5 1 
 
 2t+80+84 
 
 
 
 
 3. 2109 n 
 
 4. 3338 
 
 
 j* 
 
 2 +80+74-2 > 
 
 
 
 
 2. 068 n 
 
 3. 2092 
 
 
 
 i+50+54 
 
 
 
 9.3 fl 
 
 1. 140 
 
 2.0056 
 
 2. 5727* 
 
 * 
 
 ^+70+64 
 
 
 
 0.5 n 
 
 2. 2749 n 
 
 3. 2377 
 
 3. 9184 n 
 
 >) 
 
 |+70+74 
 
 
 
 0.3 
 
 2. 0542 
 
 3. 0565 n 
 
 3. 7710 
 
 
 ^+70+74 
 
 
 
 8.1 
 
 0. 4:) n 
 
 1.346 
 
 1. 959 B 
 
 
 (0-0 ) sin 
 
 
 
 
 . 
 
 
 
 nf 
 
 4 
 
 9.66 
 
 0. 810 B 
 
 2. 7559 
 
 3. 3840 n 
 
 3. 6946 
 
 
 r/ 
 
 20+ 4 
 
 
 9.79 
 
 0.54 
 
 
 
 
 r; 
 
 20+2J 
 
 
 9.92 
 
 0.63 n 
 
 
 
 
No. 3.] 
 
 MINOR PLANETS LEUSCHNER, GLANCY, LEVY. 
 
 TABLE C (XLIII) Continued. 
 
 Logarithmic. 
 
 33 
 
 UnJt-1" 
 
 
 (>-t?i) sin 
 
 ^ 
 
 to-* _ 
 
 
 w* 
 
 W 
 
 
 
 r 
 
 
 
 9.801, 
 
 0.425 
 
 2. 5970, 
 
 3. 1493 
 
 3.4158, 
 
 
 1* 
 
 t 
 
 1. 075, 
 
 2.045 
 
 3.5201, 
 
 4. 3751 
 
 
 
 11" 
 
 t 
 
 1.640, 
 
 2.514 
 
 4.1066, 
 
 4.8961 
 
 
 
 r>i 
 
 t 
 
 1.063 
 
 L916, 
 
 4.1066 
 
 4.8961, 
 
 5.4076 
 
 
 * 
 
 + 4 
 
 9.36 
 
 0.471, 
 
 2. 4824 
 
 3.0830, 
 
 3.3936 
 
 
 Tj*1) 
 
 *+ J 
 
 1.565 
 
 2.456, 
 
 3.9671 
 
 4.7890, 
 
 
 
 Ij" 
 
 e-f- J 
 
 1.543 
 
 2.341, 
 
 3.8942 
 
 4.6760, 
 
 
 
 f ^ 
 
 + 4 
 f+ 2J 
 
 0.87, 
 1.441, 
 
 1.75 
 2.273 
 
 4.0705, 
 3.7192, 
 
 4. 8759 
 4.5456 
 
 5.3965, 
 
 
 f *l' 
 
 + 
 
 0.42 
 
 L36, 
 
 3.7799 
 
 4. 5819, 
 
 5.0998 
 
 
 I* 7 ! 
 
 *+ J+Z 
 
 0.695, 
 
 1.585 
 
 4. 0417, 
 
 4.7827 
 
 5.2543, 
 
 
 | 
 
 t+4i>+4J 
 
 
 9.59, 
 
 0.45 
 
 
 
 
 f 
 
 t+4.+34 
 
 
 9.46 
 
 0.34, 
 
 
 
 
 1 
 
 2 f +2i+2J 
 
 
 9.45 
 
 0.11, 
 
 
 
 
 *' 
 
 2t+2,>+34 
 
 
 9.32, 
 
 0.04 
 
 
 
 
 Y 
 
 - t+ 4 
 
 L255, 
 
 2.149 
 
 3.6240, 
 
 4.4615 
 
 
 
 
 (-)< 
 
 
 
 
 
 
 
 i 
 
 , 
 
 
 9.25 
 
 0. 117, 
 
 
 
 
 '' 
 
 + -i 
 
 
 9.12, 
 
 0.02 
 
 
 
 
 
 
 m" 
 
 m" 
 
 m", m' 
 
 m' 
 
 m' 
 
 m' 
 
 
 
 j 
 
 
 
 
 
 COB Arg.+(*- l )JwijlV; 5 C', sin 
 where C,, C 2 , C, represent the respective coefficients. 
 110379 22 - 3 
 
 C, coe Aig. 
 
 
 Ml 
 
 
34 
 
 MEMOIRS NATIONAL ACADEMY OF SCIENCES. 
 
 :- TABLJB D (LIV). 
 
 2 U p . q r)Pii'V sin Arg 
 
 [Vol. XIV. 
 
 Logarithmic. 
 
 Unlt-1". 
 
 
 Sin 
 
 to-1 
 
 w* 
 
 U) 
 
 7) 
 
 - 4-n' 
 
 
 3.06^ 
 
 3. 7258 
 
 v 
 
 -n' 
 
 
 2! 8235 
 
 3. 5528 B 
 
 / 
 
 20+ 4-n' 
 
 
 2. 2831 
 
 2. 8483 n 
 
 n 
 
 40+34 -H' 
 
 1.705 
 
 3. 1591 B 
 
 3. 9166 n 
 
 V 
 
 40+24-n' 
 
 
 3. 2462 
 
 3. 8608 
 
 n 
 
 j+ -n' 
 
 
 3. 2112 B 
 
 3.8544 
 
 v 
 
 j:+ 0_|_ 4 n' 
 
 
 2. 5875 
 
 3. 4153 B 
 
 / 
 
 j+30+24-n' 
 
 
 2. 2787 
 
 2. 6304 B 
 
 7;' 
 
 if+50+34-H' 
 
 
 3.3J55 
 
 3. 5865 B 
 
 
 if +50+44 -n' 
 
 
 3. 0779 n 
 
 3. 3972 
 
 li, 
 
 -i- 0-24-n' 
 -if- 0- 4-n' 
 
 
 3. 1158 B 
 3. 1493 
 
 3. 7378 
 3. 7644 B 
 
 , 
 
 -j + -n' 
 -j +30+ 4-n^ 
 
 
 2. 3242 
 3.3863 
 
 3. 0060 B 
 4. 1833 B 
 
 T; 
 
 
 
 3. 3532 n 
 
 4. 1452 
 
 7) 
 
 +20+ 4-n' 
 
 2.6364 
 
 3. 3704B 
 
 3.8423 
 
 v 
 
 t+20+24-H' 
 
 1.423 B 
 
 2.706 
 
 3. 4014 B 
 
 
 +40+34 n' 
 
 1.4042 n 
 
 2. 1720 
 
 2. 6339 n 
 
 ^/ 
 
 +60+44 -H' 
 
 2. 3306 B 
 
 3. 1922 
 
 3. 7582 n 
 
 7) 
 
 +60+54 -H'' 
 
 2. 1137 
 
 3. 0138 B 
 
 3. 6101 
 
 
 - -20-34 -H' 
 
 2. 7175 
 
 3. 4858 B 
 
 3.9484 
 
 T/ 
 
 - -20-24-n' 
 
 2. 7756 n 
 
 3.5070 
 
 3. 9456 B 
 
 / 
 
 c 4 n' 
 
 
 1. 6810 
 
 2. 2463 B 
 
 _/ 
 
 - +20 -n' 
 
 2. 8125 
 
 3. 4427 B 
 
 3. 7846 
 
 1) 
 
 - +20+ 4-n' 
 
 2. 9121 B 
 
 3. 4958 
 
 3. 8338 
 
 7) 
 
 $+30+24 -H' 
 
 
 2.6058 
 
 3. 5312 B 
 
 v 
 
 4+30+34 n' 
 
 
 1.760 
 
 1.82 B 
 
 
 *+50+44 n' 
 
 
 1. 7510 B 
 
 2. 8113 
 
 v 
 
 $+70+54 -n' 
 
 
 2. 9120 B 
 
 4.0813 
 
 
 
 _s _30_44_n' 
 
 
 2. 8673 
 
 3. 8458 n 
 
 v 
 
 IE 30 34 n' 
 
 
 2. 9620 B 
 
 3. 9124 
 
 
 -$- 0-24-n' 
 
 
 2. 0569 B 
 
 2. 7932 
 
 7)' 
 
 if+ 4 n' 
 
 
 2. 9275 B 
 
 3. 4708 
 
 
 -je+ -n' 
 
 
 2. 9702 
 
 3. 5487 B 
 
 1} 
 
 2 +40+34-n' 
 
 
 1.640 
 
 2.7S1 B 
 
 
 2j+40+44 n' 
 
 
 1.617 
 
 2. 340 n 
 
 
 2+60+54-H' 
 
 
 1.206 B 
 
 2. 2110 
 
 7] 
 
 -2 -40-54 -n' 
 
 
 2.4012 
 
 3. 3634 B 
 
 V 
 
 -2j-40-44-n' 
 
 
 2. 5241 B 
 
 3.4544 
 
 
 -2-20-34-n' 
 
 
 1. 5290 n 
 
 2. 3210 
 
 7)' 
 
 -2e -24 -H' 
 
 
 2. 3174,, 
 
 3. 0558 
 
 
 -2 - 4-n' 
 
 
 2. 3514 
 
 3. 0737 B 
 
 
 
 tTl' 
 
 u 
 
 I COS t 
 
 =2 U P . q i)Pij'<l sin Arg.+jt JiT, (cos t- e i)+K a sin f 
 
 - i)+Cj rin t. 
 
No. 3.] 
 
 MINOR PLANETS LEUSCHNER, GLANCY, LEVY. 
 TABLE E, (LV,). 
 
 Logarithmic. K l UnJt-1'. 
 
 
 COB 
 
 P 
 
 V 
 
 
 
 1* 
 
 1 
 
 to 
 
 UUU.UUU 
 
 +++ ++ 1 
 
 ntaaaaa 
 
 2. 9180 B 
 1.9821 
 2.8036 
 3. 5175 
 3.1764* 
 3.4580 n 
 
 3.7732 
 2.5473,, 
 3.n82n 
 4. 3017, 
 3.9772 
 4. 2668 
 
 2.8138 
 
 
 m' 
 
 TABLE 
 
 t COB Arg. 
 (LV n ). 
 
 Logarithmic. 
 
 
 t COS \ 
 
 Unit-1". 
 
 
 Sin 
 
 ... 
 
 u> 
 
 * 
 
 y' 
 
 4+n' 
 
 2.9180 
 3.7799 
 1.9821* 
 3.7744* 
 3. 5175* 
 3.4580 
 3.1764 
 
 3.7732, 
 4. 5819* 
 2.5473 
 4.5420 
 4.3017 
 4.2668* 
 3.9772, 
 
 2.8138* 
 
 
 m' 
 
 sin Arg. 
 '9 ain Aig.+niT,(cos t e)+K t ein |+e,(coe e)+Cj ain 
 
 35 
 
 i sw i- 
 
 ! n O!f*.b 
 n'" 
 
36 
 
 MEMOIRS NATIONAL ACADEMY OF SCIENCES. 
 
 [Vol. XIV. 
 
 Logarithmic. 
 
 TABLE F (LVI) 
 w w a 
 
 Unit 1 radian. 
 
 
 Cos 
 
 Ml 
 
 -> 
 
 UJ-l 
 
 u-o 
 
 W 
 
 at 
 
 
 
 
 4.360 
 
 5. 1966 n 
 
 5. 7767 
 
 
 
 
 r 
 
 
 
 4.766 
 
 6. 6599 
 
 7. 3732 B 
 
 7. 7492 
 
 
 2r 
 
 
 
 4.446 
 
 7. 1194 
 
 7. 7572 B 
 
 8. 0553 
 
 
 sr 
 
 
 
 4.412 
 
 6.8442 
 
 7. 5458 B 
 
 7.9060 
 
 
 4F 
 
 
 
 4.484 
 
 6. 588,3 
 
 7. 3450 n 
 
 7.7602 
 
 
 5r 
 
 
 
 
 6. 3437 
 
 7. 1490 n 
 
 7.6136 
 
 
 7T 
 
 
 
 
 5.875 
 
 6. 7632 n 
 
 7.3134 
 
 9o 
 
 -5r+20 +2J 
 
 
 
 
 6.5090 
 
 6. 6325 n 
 
 7. 4746 B 
 
 
 -4r+20 +2J 
 -3r+20 +2J 
 
 
 
 4. Win 
 3.19 
 
 6.169 
 6. 882] 
 
 7. 0658 
 7. 6078 
 
 7. 86980, 
 7. 9975 B 
 
 
 -2r+20 +2J 
 
 
 
 3.52 
 
 7. 098fi n 
 
 7. 6970 
 
 7. 9394 n 
 
 
 - r+20 +2J 
 
 
 
 5. 1420 
 
 6.359 
 
 7. 0722 n 
 
 7.4480 
 
 
 20 +2J 
 
 
 4.379 
 
 7. 6355 B 
 
 8. 2144 
 
 8. 4125 B 
 
 
 
 r +20 +2J 
 
 
 
 4. 856 n 
 
 8. 0894 B 
 
 8.9548 
 
 9. 5668 B 
 
 
 2r+20 +2J 
 
 
 C A 
 
 4.92 
 
 7. 8150 n 
 
 8. 6561 
 
 9. 2006 B 
 
 
 3r+20 +2J 
 
 
 
 5. 5174, 
 
 7. 6056 n 
 
 8.4650 
 
 9. 0111 B 
 
 
 4r+20 +2J 
 
 
 
 5. 4248 n 
 
 7.4128 B 
 
 8. 2958 
 
 8. 8561 B 
 
 
 5r+20 +2J 
 
 
 
 
 7.2254, 
 
 8. 1426 
 
 8. 7346 B 
 
 
 7r+20 +2J 
 
 
 
 
 6. 8746 B 
 
 7. 8484 
 
 8. 4936 B 
 
 J 
 
 -5r+20 + J 
 
 
 I^rf i 
 
 
 6. 8776,, 
 
 7.5604 
 
 7. 8425 B 
 
 
 4f +20 + J 
 
 
 ! _tvjfl? .1 
 
 4.582 
 
 6. 8815 n 
 
 7.4536 
 
 7. 5238^ 
 
 
 -3r+20 + J 
 
 
 M'*T . 
 
 4.674 
 
 6. 6271 B 
 
 6. 7816 
 
 7. 3174 
 
 
 -2r+20 + J 
 
 
 n<3ri<5 .il 
 
 4.99 
 
 6. 7985 
 
 7. 4732 n 
 
 7.7966 
 
 
 - r+20 + J 
 
 
 
 5. 4623 B 
 
 
 
 
 
 20 + Jo 
 
 
 4.605 B 
 
 7. 1987 
 
 7. 8314 B 
 
 8. 1061 
 
 
 
 r+20 + J 
 2r+20 + J 
 
 
 
 5.0056 
 4.38 
 
 8. 2964 
 8. 0434 
 
 9. 1086 B 
 8. 831 6 B 
 
 9.6833 
 9. 3296 
 
 
 sr+20 + J 
 
 
 
 5. 6251 
 
 7. 8458 
 
 8. 6564 n 
 
 9.1558 
 
 
 4r+20 + J 
 
 
 
 5.5812 
 
 7.6603 
 
 8. 5030 n 
 
 9. 0248 
 
 
 5r+20 + J 
 
 
 1 IH t-it>' v .' 1* 
 
 
 7. 4778 
 
 8. 3544 n 
 
 8.9050 
 
 
 7r+20 + J 
 
 
 " "* t S v ( 
 
 
 7. 1130 
 
 8. 0545 B 
 
 8. 6668 
 
 I* 
 
 n t -.f( -1>T>, 
 
 4.664 
 
 4.71 
 
 5.83 
 
 
 
 
 
 r 
 
 
 
 
 7. 8102 
 
 8. 6250 B 
 
 
 
 2r 
 
 
 
 
 7. 7520 n 
 
 8. 1242 
 
 
 
 sr 
 
 
 
 
 7. 6172 B 
 
 6. 6043 B 
 
 
 
 4r 
 
 
 
 
 7. 7135 n 
 
 8.2308 
 
 
 9o* 
 
 _4r+40 +4J 
 
 
 
 
 7. 1862 
 
 7. 9072 B 
 
 
 
 3r"-(-40o+4J 
 
 
 
 
 7.1804 
 
 7. 8679 B 
 
 
 
 -2r+40 +4J 
 
 
 
 
 6.817 
 
 7. 456 n 
 
 
 
 r"+40 +4Jo 
 
 
 
 
 8. 4680 n 
 
 8. 8822 
 
 
 
 40 -f-4J 
 
 4.666 
 
 5. 807 B 
 
 8. 0913 
 
 8. 8270 n 
 
 9.2073 
 
 
 
 pjf^g +4J 
 
 
 
 
 8.7850 
 
 9. 8236 n 
 
 
 
 2T +40 +4Jo 
 
 
 
 
 8. 5144 
 
 9. 4910 n 
 
 
 
 3/ 1 +40o+4Jo 
 
 
 
 
 8. 3274 
 
 9. 3006 B 
 
 
 
 4r+40 +4J 
 
 
 
 
 8. 1627 
 
 9. 1494 n 
 
 
 
 57"*-f-4vQ~}~4^o 
 
 
 
 
 8. 0050 
 
 9. 0105 n 
 
 
 in' 
 
 4/~'-t-40 -4-3^ 
 
 
 
 
 7.354 n 
 
 8. ]083 
 
 
 
 -3^+40 +3J 
 
 
 
 
 7. 5708 n 
 
 8. 2084 
 
 
 
 ~~ ^ f ~f~40Q-f~3Jo 
 
 
 
 
 8.8838 
 
 9.0548- 
 
 
 
 40 -f-3Jo 
 
 4. 516 n 
 
 6. 2084 
 
 8. 5565 B 
 
 9. 218p 
 
 9. 5174 n 
 
 
 
 / 1 +40 +3J 
 
 
 
 
 9. 2783 B 
 
 0. 2833 
 
 
 
 2/^+400 +3J 
 
 
 
 
 9. 0241n 
 
 9. 9635 
 
 
 
 3r+40 +3J 
 
 
 
 
 8. 8480 n 
 
 9.7850 
 
 
 
 4/^+400 +3J 
 
 
 
 
 8. 6916 B 
 
 9.6434 
 
 
 
 5r+40 +3J 
 
 
 
 
 8.5401,, 
 
 a 5128 
 
 
 
 
 m/a 
 
 m/J 
 
 77l /2 , 77J- 7 
 
 m'\ m' 
 
 m' 
 
 m' 
 
No. 3.] 
 
 MINOR PLANETS LEUSCHNER, GLANCY, LEVY. 
 
 TABLE F (L VI) Continued. 
 
 37 
 
 Logarithmic. 
 
 w I 
 
 Unit- 1 radian. 
 
 
 Cos 
 
 ^ 
 
 w-J 
 
 ..; 
 
 10 
 
 . 
 
 TUB' 
 
 -4T ~ + ^o 
 
 
 
 
 7.7640 
 
 7.8364, 
 
 
 /w / 
 
 -sr f 4 
 
 
 
 
 7.4203 
 
 8. 3915 
 
 
 
 
 
 
 
 7. 8104, 
 
 8.6268 
 
 
 
 - r f- J, 
 
 
 
 
 8. 0479, 
 
 8.8018 
 
 
 
 
 4.518, 
 
 5.886, 
 
 5.70, 
 
 
 
 
 
 r + 
 
 
 
 
 7. 1339 
 
 7.8500 
 
 
 
 
 
 
 
 7.8421 
 
 a 4293, 
 
 
 
 sr + A 
 
 
 
 
 7.9669 
 
 a 6796, 
 
 
 
 4T + Jo 
 
 
 
 
 7.9760 
 
 a 7576, 
 
 
 *" 
 
 _4r+40 +2J 
 
 
 
 
 6.9002 
 
 7.6938, 
 
 
 
 -3r+4^o+2J 
 
 
 
 
 7.1638 
 
 7.8502, 
 
 
 
 I 2 f+4J7+2J 
 
 
 
 
 8.1860, 
 
 a 4016 
 
 
 
 49 +2^ 
 
 3.76 
 
 e-oeoSn 
 
 8. 4157 
 
 8. 9760, 
 
 9. 1661 
 
 
 
 /'+4^o+2J 
 
 
 
 
 9.1714 
 
 0.1382, 
 
 
 
 2f-(-4fl -)-2J- , 
 
 
 
 
 8.9358 
 
 9.8333, 
 
 
 
 3y-j_4# -j-2J 
 
 
 
 
 8. 7718 
 
 9. 6681 B 
 
 
 
 4/^-j-4^ -|-2^ 
 
 
 
 
 8.6236 
 
 9.5372, 
 
 
 
 
 
 
 
 ***' 
 
 
 
 j.n 
 
 
 176 
 
 5. 7516 
 
 4.7 
 
 
 
 
 
 r 
 
 
 
 
 7.8677 
 
 8. 6727, 
 
 
 
 2f 
 
 
 
 
 7. 8610, 
 
 8.2228 
 
 
 
 sr 
 
 
 
 
 8. 1026, 
 
 8.7296 
 
 
 
 4r 
 
 
 
 
 8.1538, 
 
 8.8728 
 
 
 f 
 
 r 
 
 
 
 
 7.9418, 
 
 a 7337 
 
 
 
 2f 
 
 
 
 
 7.9312, 
 
 a 7154 
 
 
 
 sr 
 
 
 
 
 7.7920, 
 
 8.6154 
 
 
 
 4r 
 
 
 
 
 7.639, 
 
 a 5001 
 
 
 f ;.' ' :: 
 
 
 
 
 
 
 
 
 f 
 
 _4f-(-4fl -)-3 < / _j 
 
 
 
 
 7.446 
 
 8.1156, 
 
 
 
 -Sr+^o+SJo-J',, 
 
 
 
 
 7.1858 
 
 7. 8677 B 
 
 
 
 - r+4+3Jo-^o 
 
 
 
 
 7. 6176, 
 
 7.9693 
 
 
 
 4^ -|-3J 2 a 
 
 
 4.804, 
 
 7.168 
 
 7.9368, 
 
 a 3724 
 
 
 
 /-(-4^ -j.3J ^" 
 
 
 
 
 7.7887 
 
 8.8492 B 
 
 
 
 2r +4^ -j-3J .J 
 
 
 
 
 7.448 
 
 a 4531, 
 
 
 
 3r-j-4tf -i-3J ^ 
 
 
 
 
 7. 19J6 
 
 8.2026, 
 
 
 
 4r +4^o+3J -^o 
 
 
 
 
 6.978 
 
 7.9963, 
 
 
 ** 
 
 2^ +2J 
 
 5.418, 
 
 6.292 
 
 7. 4754, 
 
 a 6636 
 
 
 
 
 6/? +6J 
 
 5. 418, 
 
 6.292 
 
 8.6328, 
 
 9.4351 
 
 
 
 1J0 2 !/ 
 
 26 + ^ 
 
 5.885 
 
 6. 719, 
 
 8.5059 
 
 9- 2804, 
 
 
 
 wV 
 
 2C +3J 
 
 4.974 
 
 5.896, 
 
 8. 0326, 
 
 8^ 1975 
 
 
 
 r ">X 
 
 6 +5J 
 
 5.935 
 
 6.780, 
 
 9. 2774 
 
 0.0330, 
 
 
 
 
 2^o 
 
 5.744, 
 
 6.535 
 
 8. 3811, 
 
 9.1030 
 
 
 
 r /0 j;' 2 
 
 20 +2J 
 
 5.441, 
 
 6.327 
 
 a 0917 
 
 8.6300 
 
 
 
 'So 'j' 2 
 
 66 +4J 
 
 5. 919., 
 
 6.744 
 
 9.4432, 
 
 0.1464 
 
 
 
 1 ! /3 
 
 2^o+ ^o 
 
 5.301 
 
 6.14.9, 
 
 8.2302 
 
 9. 0152, 
 
 
 
 '" 
 
 6C +3J 
 
 5.301 
 
 6. 149, 
 
 9.1294 
 
 9. 7729, 
 
 
 
 
 2$ +2J 
 
 
 
 8.5904 
 
 9.3492, 
 
 9.8022 
 
 
 /'Jo 
 
 2^ + J 2<t 
 
 4.502 n 
 
 5.41 
 
 a 1011, 
 
 8.8726 
 
 
 
 
 6^ +5J ^" 
 
 4.502, 
 
 5.41 
 
 8.0554, 
 
 8.9263 
 
 
 
 *3 _/ 
 J i 
 
 2^ + J 
 
 
 
 8.5592, 
 
 9. 3245 
 
 
 
 f i 
 
 26 +2J 2 a 
 
 4.057 
 
 5.021, 
 
 6.887 
 
 8.1804, 
 
 
 
 f i 
 
 6<? +4J -v 
 
 4.057 
 
 5.021, 
 
 8. 2718 
 
 9. 1021, 
 
 
 
 ^'9j^ coe Arg. 
 
 where C represents the coefficient. 
 
MEMOIRS NATIONAL ACADEMY OF SCIENCES. 
 
 [Vol. XIV. 
 
 Logarithmic. 
 
 TABLB G (LVII). 
 S sin </>+C coa if/ 
 
 Unlt-1". 
 
 
 Cos 
 
 to- 
 
 te-* 
 
 M 
 
 . 
 
 1C 
 
 . 
 
 
 <1> 5r+20 +2 J 
 
 
 
 8.81 
 
 1.082 B 
 
 1. 5710 
 
 1. 612 B 
 
 
 <l> 4f +20 +2J 
 
 
 
 9.009 
 
 
 1.5493 
 
 0. 9,89 
 
 
 <j> 3/ > +20o+2J 
 
 
 
 9.318 
 
 0^931 
 
 1. 604 B 
 
 1.916 
 
 
 <A 2.r+20o+2J 8 
 
 
 
 9.207 
 
 1. 6478 
 
 2. 1070 B 
 
 2. 2333 
 
 
 <fr- r+20 +2J 
 
 
 
 9.711 
 
 1.950 
 
 2. 3426 B 
 
 2. 3713 
 
 
 <& +20 +2J 
 
 
 9.196 
 
 2. 171 2 n 
 
 2. 5678 
 
 2. 565 B 
 
 
 
 
 
 
 9. 230 n 
 
 2. 3541 n 
 
 3. 1493 
 
 3. 7107 B 
 
 
 ^+27" I +20 +2J 
 
 
 
 9. 220 n 
 
 1. 9114 n 
 
 2. 6867 
 
 3. 1657 n 
 
 
 ^+3r+20 +2J 
 ^+4/ I +20 +2J 
 
 
 
 a494n 
 
 1. 5372 B 
 1.2544,, 
 
 2. 3831 
 2. 1315 
 
 2. 8623 B 
 2. 6333 B 
 
 
 ^+5/ 1 +20 +2J 
 
 
 
 9.100 B 
 
 1.018 B 
 
 1.9034 
 
 2. 4248 n 
 
 to 
 
 ^-5r+40 +4J 
 
 
 
 9.771 B 
 
 1.042 B 
 
 .1. 868 
 
 2. 357 B 
 
 
 d> 4/'+40 +4J 
 
 
 
 0. O64. n 
 
 1. 723 n 
 
 2. 3515 
 
 2. 6814 n 
 
 
 ^ 3r+40 +4J 
 
 
 
 0. 3185 n 
 
 2. 1626 n 
 
 2. 6961 
 
 2. 921 4 B 
 
 
 tj 2/ 1 +40 +4Jo 
 
 
 
 0. 497 B 
 
 2. 7787 B 
 
 3.0649 
 
 3. 0993,, 
 
 
 #- r+40 +4J 
 
 
 
 1. 0286 
 
 3. 2379 n 
 
 3. 1223 
 
 3. 9385 B 
 
 
 # +40 +4J 
 
 9.199 
 
 9.04 B 
 
 2. 6172 
 
 3. 2511 n 
 
 3. 4930 
 
 
 
 0+ r+40 +4J 8 
 
 
 
 0.7226 
 
 3. 1702 
 
 4. 1580 B 
 
 4. 9365 
 
 
 <&+2r+40 +4J 8 
 
 
 
 0.669 
 
 2. 7877 
 
 3. 7083 B 
 
 4. 3605 
 
 
 i+3f +40 +4J 
 
 
 
 0.9435 
 
 2. 5117 
 
 3. 426L, 
 
 4.0450 
 
 
 ^+4r+40 +4J 
 
 ~ j 
 
 
 0. 5122 
 
 2. 2732 
 
 3. 2042 B 
 
 
 to 
 
 <fi-5r 
 
 
 
 9.814 
 
 1.925 
 
 2. 634 
 
 2.984 
 
 
 <!>-r 
 
 
 
 0. 0434 B 
 
 2. 0527 
 
 2. 6896 B 
 
 2. 9432 
 
 
 <bsr 
 
 
 
 0. 3541 B 
 
 2.145 
 
 2. 675 B 
 
 2.744 
 
 
 d> 2/ 1 
 
 
 9.140 
 
 0. 362 B 
 
 2. 1351 
 
 2. 3850 B 
 
 2.4864n 
 
 
 <j> F 
 
 
 
 0. 4164 n 
 
 2.3504 B 
 
 3. 0929 
 
 3. 5397 B 
 
 
 | 
 
 
 9. 274 n 
 
 0. 1436 B 
 
 
 
 
 
 
 
 
 0. 3102 n 
 
 2.497 
 
 3. 1875 B 
 
 3. 5978 
 
 
 ^+2.T 
 
 
 9. 137 B 
 
 9.918 
 
 1.9006,, 
 
 1.0453 
 
 2.8834 
 
 
 J+ttf" 
 
 
 
 9.465 
 
 0. 8 / 12 n 
 
 2. 5218 B 
 
 3. 3564 
 
 
 ^+4jT 
 
 
 
 9.20 n 
 
 1.406 n 
 
 1.729 
 
 
 if 
 
 5r"+45o+3Jo 
 
 
 
 9.476 
 
 1.327 
 
 L889 B 
 
 2. 2299 
 
 
 4/ I +40 +3J 
 
 
 
 9.781 
 
 1.447 
 
 2. 1506 B 
 
 2.5419 
 
 
 ^ 3.T+40 +3J 
 
 
 
 9.811 
 
 2. 1070 
 
 2. 6309 n 
 
 2. 8J508 
 
 
 ^ 2.T+40 +3 J 
 
 
 
 0.3489 
 
 2. 5095 
 
 2. 9557 B 
 
 3. 0952 
 
 
 <]>- r+40 +3J 8 
 
 
 
 0. 9511 
 
 3. 3599 
 
 2. 7758 
 
 3. 9726 
 
 
 <!> +40 +3J 
 
 8.76 B 
 
 0.158 
 
 2. 7932 B 
 
 3. 3085 
 
 3. 4526 B 
 
 
 
 </>+ r+40 +3J 
 
 
 
 9. 961,, 
 
 3. 3609 n 
 
 4.3114 
 
 5. 0691 n 
 
 
 ^+2/'+40 +3J 8 
 
 
 
 491 
 
 2. 9943,, 
 
 3. 8728 
 
 4. 4922 B 
 
 
 ^-j-3/ 1 +40o+3J 
 
 
 
 1.' 0464 B 
 
 2. 7293,, 
 
 3.6067 
 
 4. 1945 B 
 
 
 ^+4f+40 +3J 
 
 
 
 0. 678 n 
 
 2. 4992 n 
 
 3. 3946 
 
 
 Tf 
 
 V ~~ 5j ~T~ Jn 
 
 
 
 9.848 
 
 2. 0766 n 
 
 2.712 
 
 2. 9697 n 
 
 
 V^4l "4~ Jn 
 
 
 
 0. 0792 
 
 2. 1609 n 
 
 2. 6968 
 
 2. 7976 B 
 
 
 <j>zr + J 
 
 
 
 0.3941 
 
 2. 157 n 
 
 2.491 
 
 1.51 
 
 
 <j>zr + J 
 
 
 9. 013 n 
 
 0.248 
 
 2. 0455 
 
 2. 7898^ 
 
 3. 2380 
 
 
 <ii r + Jo 
 
 
 
 9.901 
 
 2. 58,4 
 
 3. 2539 n 
 
 3. 6434 
 
 
 1 
 
 
 9. 885 n 
 
 0. 8,518 
 
 
 
 
 
 ^+ /" + Jo 
 
 
 
 0.1664 
 
 1.836 
 
 2.448 
 
 3. 3029 B 
 
 
 ^+2r + J 
 
 
 9.009 
 
 9.76 B 
 
 2. 1633 
 
 2. 6170 n 
 
 2. 2433 
 
 
 ^+sr + J 
 
 
 
 9.38 B 
 
 2. 1064 
 
 2. 7194 B 
 
 2. 9212 
 
 
 ^+4r + J 
 
 
 
 
 1. 9892 
 
 2. 6870 B 
 
 
 V 
 
 ^-5r+60 +6J 
 
 
 
 
 2.3144 
 
 2. 9730 B 
 
 
 
 <l> 4/ 1 +60 +6J 
 
 
 
 
 2. 9538 
 
 3. 3785 n 
 
 
 
 ^ 3r t +60 +6J 
 
 
 
 
 3. 3102 
 
 3. 5843 n 
 
 
 
 y~2/ 1 +60 +6J 
 
 
 
 
 3. 4970 
 
 3. 8423 B 
 
 
 
 ^ ^+60o+6J 
 
 
 
 
 3.9455 
 
 3. 7269 n 
 
 
 
 ^ +60 +6J 
 
 9.95 n 
 
 1. 1109 n 
 
 3. 1673 B 
 
 3. 9296 
 
 4. 3377,, 
 
 
 
 Y~\~ ^ 1 +60o+6J 
 
 
 
 
 3. 9144 n 
 
 5. 0372 
 
 
 
 ^+2f+60o+6J 
 
 
 
 
 3. 5594 B 
 
 4. 5942 
 
 
 
 ^+3r+60 +6J 
 
 
 
 
 3. 3121 B 
 
 4. 3236 
 
 
 to 3 
 
 ^-5r+20 +2J 
 
 
 
 
 2. 1657 
 
 2. 7221 B 
 
 
 
 d> 4/ 1 +20 +2J 
 
 
 
 
 2. 1255 
 
 2. 8004 B 
 
 
 
 ^ 3r+20o+2J 
 
 
 
 
 2.234 
 
 3. 1304 n 
 
 
 
 <!> 2r+20 +2Jo 
 
 
 
 
 2.576 
 
 3. 3804 n 
 
 
 
 <1> ^"+200+2^0 
 
 
 
 
 3. 1995 
 
 3. 8325 n 
 
 
 
 ip +20 +2J 
 
 0. 344 n 
 
 1.017 
 
 2. 689 n 
 
 3. 4822 
 
 3. 9938 n 
 
 
 
 <l>-\- / 1 +20o+2J 
 
 
 
 
 2. 2480 
 
 3. 2839 n 
 
 
 
 ^+2r+20 +2J 
 
 
 9.45 
 
 
 3. 1612 
 
 3. 8424 n 
 
 
No. 8.] 
 
 MINOR PLANETS LEUSCHNER, GLANCY, LEVY. 
 
 39 
 
 Logarithmic. 
 
 G (LVII) Continued. 
 S sin <j>+C coe <j> 
 
 Unlt-l". 
 
 
 Cos 
 
 ^ 
 
 ^ 
 
 ,H 
 
 ; 
 
 1C 
 
 - 
 
 , 
 
 f_5r-20 -2J 
 
 
 
 
 2.700, 
 
 15481 
 
 
 TO 
 
 \_ip 20 2J 
 
 
 
 
 2. 817, 
 
 3. 6251 
 
 
 
 V_or- 20 2J 
 
 
 
 
 2. 9247, 
 
 3.6905 
 
 
 
 i 2.T 20 2J 
 
 
 9.59, 
 
 
 3. 0241 B 
 
 17470 
 
 
 
 <i F 20 2J 
 
 
 
 
 3. 1364, 
 
 3.8346 
 
 
 
 i 20 24 
 
 0. 117 0. 95, 
 
 2.297, 
 
 2.7856, 
 
 3.6614 
 
 
 
 i+ F 20 2A 
 
 
 
 2.8942, 
 
 3.5604 
 
 
 
 (4+2.T 20 2J 
 
 
 
 
 2.297, 
 
 11129 
 
 
 
 
 
 
 
 
 
 
 if 
 
 <i 5F+60 +5J 
 
 
 
 
 
 2.4885, 
 
 11691 
 
 
 vo T 
 
 <A 4/ f +60+5J 
 
 
 
 
 2.976, 
 
 15560 
 
 
 
 5 3r+60 +5J 
 
 
 
 
 3.6541, 
 
 18829 
 
 
 
 (4 2/"+60 +5J 
 
 
 
 
 3.9514, 
 
 4.1632 
 
 
 
 /^+60A+5Jg 
 
 
 
 
 4.3903, 
 
 4.0037, 
 
 
 
 L'' +60/\+5Jo 
 
 0.295 
 
 1.366 
 
 3.6364 
 
 4.3301^ 
 
 4.6662 
 
 
 
 $+ f+60 +5J 
 
 
 
 
 4.4005 
 
 5.4966, 
 
 
 
 <4+2/ 1 +60 +5J 
 
 
 
 
 4.0582 
 
 5.0612, 
 
 
 
 ^+3r+60+5Jo 
 
 
 
 
 3.8204 
 
 4.8027, 
 
 
 . 
 
 v 5 "T 2^o < ^o 
 
 
 
 
 2.426, 
 
 10684 
 
 
 
 ^ 4/^_i_2^ 4-J 
 
 
 
 
 2. 399, 
 
 3. 0310 
 
 
 
 ^ 3/^-4-2$ +^ 
 
 
 
 
 2. 410, 
 
 3.1305 
 
 
 
 5--2.T-i-20 +J 
 
 
 
 
 2.701, 
 
 14602 
 
 
 
 y~~ ' i 2vQ~T~ o 
 
 
 
 
 3.2842, 
 
 3.8558 
 
 
 
 V *i *^o~i o 
 
 0.444 
 
 1.188, 
 
 3.0569 
 
 3,7266, 
 
 4.1122 
 
 
 
 > i r^ I Oa i ^ 
 
 
 
 
 2.8541 
 
 3.5823, 
 
 
 
 ^+2r+2flJ+4 
 
 
 
 
 3. 2191, 
 
 17635 
 
 
 , 
 
 A-5r-20 -J a 
 
 
 
 
 3.1551 
 
 19530, 
 
 
 
 <l>4r26 J 
 
 
 
 
 3.2454 
 
 3.9948, 
 
 
 
 A 3f 20. 4. 
 
 
 
 
 3.3100 
 
 4.0023, 
 
 
 
 ibir 20 j 
 
 
 9.93 
 
 
 3.3277 
 
 3.9401, 
 
 
 
 ij /" 20 J 
 
 
 
 
 3. 1976 
 
 3. 4598, 
 
 
 
 A 20 J 
 
 0.490, 
 
 1.324 
 
 10145, 
 
 3.7326 
 
 4.2787 
 
 
 
 i+ f 20 J 
 
 
 
 
 3.3632 
 
 3.9402, 
 
 
 
 ^+2r-20 -j 
 
 
 
 
 2.7792 
 
 15224, 
 
 
 fcf / 
 
 ,j_5/-+20 +34, 
 
 
 
 
 2.2738, 
 
 2.847 
 
 
 
 V 1 ~~"4j ~|~Urt~T~"^O 
 
 
 
 
 2.116, 
 
 3.0290 
 
 
 
 tj 3/^+2^0 -j- 3^ 
 
 
 
 
 2.5858, 
 
 3. 3787 
 
 
 
 V ~~2/ ~y~toVft~7~O^Q 
 
 
 
 
 2.809, 
 
 3.5429 
 
 
 
 A / ? -|-2^ -i-3J 
 
 
 
 
 2.650, 
 
 17297 
 
 
 
 W ~T~"0~1 "^0 
 
 9.98 
 
 0. 60 n 
 
 2.873, 
 
 aess 
 
 17980 
 
 
 
 cj+ /'+20 +3J 
 
 
 
 
 3.5126, 
 
 4.2856 
 
 
 
 #+2r+20 +3J 
 
 
 9.46, 
 
 
 13438, 
 
 4.1208 
 
 
 ,/J 
 
 w~~df ~f~Ov/*T~4d() 
 
 
 
 
 L9950 
 
 2.7422, 
 
 
 
 ~"~4y ~4~6w(|~j~4dn 
 
 
 
 
 2.6112 
 
 3. 1949, 
 
 
 
 d 3.T +60 +4J 
 
 
 
 
 3.0556 
 
 15583, 
 
 
 
 tf 2f+60 +4J 
 
 
 
 
 17934 
 
 17947, 
 
 
 
 ^ / I +60 +4Jo 
 
 
 
 
 4.2260 
 
 4.4064 
 
 
 
 ^ ^-60 +4J 
 
 9.98, 
 
 0.76, 
 
 3.5017, 
 
 4.1098 
 
 4.3552, 
 
 
 
 ^+ /'+60 +4J 
 
 
 
 
 4.2852, 
 
 5.3521 
 
 
 
 #+2r+60 +4J 
 
 
 
 
 3.9567, 
 
 4.9249 
 
 
 * 
 
 tJ-5r+20 e +2J 
 
 
 
 
 2.5018 
 
 10963, 
 
 
 
 # 4/'+20 +2J 
 
 
 
 
 2.453 
 
 10935, 
 
 
 
 d> 3/"+20 +2J 
 
 
 
 
 2.4799 
 
 3. 2779, 
 
 
 
 ^ 2/'+20 +2J|) 
 
 
 
 
 a 9375 
 
 16294, 
 
 
 
 ^ r"+20 +2J 
 
 
 
 
 12833 
 
 3.8982, 
 
 
 
 d> +20 +2J 
 
 0.025 B 
 
 0.60 
 
 2.634 
 
 3. 2781 
 
 4.0439, 
 
 
 
 $~T~ * t~2vn~\~2an 
 
 
 
 
 3.5607 
 
 4.2381, 
 
 
 
 ^+2r+20 +2J 
 
 
 
 
 14629 
 
 4.1704, 
 
 
 1* 
 
 ^_5r-2 
 
 
 
 
 3.0090,, 
 
 1 7477 
 
 
 
 ^ 4f 20 
 
 
 
 
 3.0676, 
 
 17445 
 
 
 
 <f>zr2o a 
 
 
 
 
 3. 0764, 
 
 3.6664 
 
 
 
 A 2r28 
 
 
 
 
 2. 958-, 
 
 1 3121 
 
 
 
 <j> r26 
 
 
 
 
 3. 1140 
 
 4.0201, 
 
 
 
 r -20 
 
 0. 305 L 127, 
 
 2.912 
 
 3.5491, 
 
 3.9085 
 
 
 
 A-\- P 26 a 
 
 I 
 
 
 3. 0396, 
 
 3.6320 
 
 
 
 A+2F-26, 
 
 
 
 2.4706, 
 
 12330 
 
 
40 
 
 MEMOIRS NATIONAL ACADEMY OF SCIENCES. 
 
 TABLE G (LVII) Continued. 
 
 Logarithmic. 
 
 [Vol. XIV. 
 
 Unit-l". 
 
 
 Cos 
 
 tu- 
 
 w-J 
 
 w-> 
 
 u,o 
 
 to 
 
 w> 
 
 f 
 
 ^-5r+66' l0 +5^ -J' 
 
 
 
 
 2.006 
 
 2. 7505 n 
 
 
 
 ^-4r+60 +5^ -j 
 
 
 
 
 2.335 
 
 2. 981, 
 
 
 
 V 3r+60 +5<l -.r o 
 
 
 
 
 2.544 
 
 3. 1436 B 
 
 
 
 <!- 2r+60 +5J -. 
 
 
 
 
 2.718 
 
 3. 2445 n 
 
 
 
 <l>- r+60 +5J -.T 
 
 
 
 
 2.970 
 
 2. 911 6 B 
 
 
 
 V* +60 +5J -.T 
 
 8.6 n 
 
 9.7 
 
 2- 1H B 
 
 2.9.23 
 
 3. 4067, n 
 
 
 
 V>+ r+60 +5^ -J 
 
 
 
 
 2. 7948 n 
 
 3. 9420 
 
 
 
 0+2r+60 +5J -.r o 
 
 
 
 
 2. 3824 n 
 
 3. 4488 
 
 
 f 
 
 ,H5r+20 +24, 
 
 
 
 
 9,6 
 
 2.387 
 
 
 
 ^-4r+20 +24 
 
 
 
 
 1. 916, 
 
 2.911 
 
 
 
 v !'-3,r+20 +24 ) 
 
 
 
 
 2. 5178 n 
 
 3.3047 
 
 
 
 0-2T+200+24, 
 
 
 
 
 2. 938 n 
 
 3. 6294 
 
 
 
 ^- r+20 +2J 
 
 
 
 
 3. 3406 n 
 
 3. 9330 
 
 
 
 ^ +2S +2^ 
 
 
 0. 5910 
 
 3. 1266 
 
 3. 8021 
 
 4. 1894 
 
 
 
 ^+ r+20 +2J 
 
 
 
 
 3. 4070 
 
 4. 3178 B 
 
 
 
 ^+2r+20 +2^ 
 
 
 
 
 3. 0472 
 
 3. 9308 B 
 
 
 JT 
 
 ^-5r-20 -4+.r o 
 
 
 
 
 0. 732 n 
 
 1.085 
 
 
 
 0-4r-20 -4,+.J 
 
 
 
 
 0.35 
 
 1. 895 n 
 
 
 
 ^-3r-2e -4,+j 
 
 
 
 
 1.463 
 
 2. 5146 B 
 
 
 
 ^-2r-25 -J +2 > 
 
 
 
 
 2.064 
 
 3. 0255 n 
 
 
 
 V r-29 -j +j 
 
 
 
 
 2. 6816 
 
 3. 6280 n 
 
 
 
 ^ -26 -J +2 
 
 9.04 
 
 o.n n 
 
 2.636 
 
 3. 3284 n 
 
 3. 7399 
 
 
 
 0+ r-2e -j +^o 
 
 
 
 
 3. 0572 n 
 
 3. 6430 
 
 
 
 ^+2r-2ff -J +^ 
 
 
 
 
 2. 9121 n 
 
 3.5491 
 
 
 V. 
 
 ^+ 40 +44 
 
 0. 775 
 
 1.65 n 
 
 3. 1052 n 
 
 3. 0342 n 
 
 
 
 
 <j>- 40 -4J 
 
 0.2ft, 
 
 1.10 
 
 3. 1888 
 
 3. 6104 n 
 
 
 
 
 ^+ 80 +8J 
 
 0.65 
 
 1.54 n 
 
 3.7520 
 
 4. 5812 
 
 
 
 *V 
 
 <P+ 40 +54 
 
 
 
 3. 7577 
 
 4. 3244 n 
 
 
 
 
 ^+ 40 +3J 
 
 1. 260 n 
 
 2.081 
 
 3. 1240 
 
 4. 1388 
 
 
 
 
 ^- 400-34, 
 
 1.005 
 
 1.77 n 
 
 3. 5356 n 
 
 3. 3560 
 
 
 
 
 0+ 8 +7J 
 
 1.228 
 
 2.093 
 
 4. 3980 n 
 
 5. 1827 
 
 
 
 W 
 
 ^+ 40 +4J 
 
 
 
 4. 1155 
 
 4.5547 
 
 
 
 
 ^+ 40 +2J 
 
 1.106 
 
 1.88 B 
 
 2.831 
 
 4. 1803 n 
 
 
 
 
 ^- 40 -2J 
 
 1. 146 n 
 
 1.88 
 
 3. 0422 
 
 4.0180 
 
 
 
 
 v '.+ 80 +6J 
 
 1.321 
 
 2. 152 n 
 
 4. 5658 
 
 5. 3010 n 
 
 
 
 v 
 
 0+ 45 +3A 
 
 
 
 3. 8375 
 
 4. 0446 n 
 
 
 
 
 #- 40 - 4, 
 
 
 
 3. 2197 
 
 3. 9650 n 
 
 
 
 
 VH- 80 +5J 
 
 
 
 4. 2553 n 
 
 4. 9349 
 
 
 
 
 
 
 
 
 
 
 
 ? 7o 
 
 V>+ 40,+34 -^ 
 
 
 
 3.0024 
 
 3. 8634 n 
 
 
 
 
 <f>- 400-340+2 1 ,, 
 
 9.98 
 
 0.8 
 
 2. 956 n 
 
 3. 8331 
 
 
 
 
 ^+ 80 +7J -J 
 
 
 
 3. 0757 
 
 3. 975? n 
 
 
 
 
 </>+ 40 +4J 
 
 0.46 n 
 
 1.32 
 
 3. 8514 n 
 
 4. 6436 
 
 
 
 ? 1' 
 
 0+ 40 +4J -J 
 
 
 
 2.442 
 
 1. 846 B 
 
 
 
 
 #- 40 -2J +2- 
 
 
 
 3. 2486 
 
 4. 0585 n 
 
 
 
 
 ^+ 80 +6J -J 
 
 
 
 3. 2818 n 
 
 4. 1441 
 
 
 
 
 ^+ 40 +3J 
 
 0.27 
 
 1.15* 
 
 3.9421 
 
 4. 6972 B 
 
 
 
 S sin t+C cos ^='SCw*riPri'Qj 2 t cos Arg. 
 where C represents the coefficient. 
 
H. TABLES FOR THE DETERMINATION OF THE PERTURBATIONS OF THE 
 
 HECUBA GROUP OF MINOR PLANETS. 
 
 DEVELOPMENT OF THE DIFFERENTIAL EQUATIONS FOR W AND FOR THE THIRD COORDINATE. 
 
 It would be futile to attempt to give a brief but comprehensive outline of the fundamental 
 developments in the theory of Bohlin-v. Zeipel which would assist the reader to an understanding 
 of the construction of the tables. In broad outlines, the problem is the integration of Hansen's 
 
 differential equations for nSz, v, and -> by means of the method developed by Bohlin and 
 
 according to the modifications introduced by v. Zeipel for purposes of numerical computation. 
 The first division of the problem is the development of functions of the partial derivatives of 
 the perturbative function; the second division of the problem is the integration of the Hansen 
 equations in the form of infinite series. 
 
 For the theory the reader is referred to the original works of Hansen 1 , Bohlin 2 , and v. Zeipel*. 
 As indicated in the introduction to the first section, unless otherwise stated, the references to 
 Bohlin refer to the French edition and are designated by B; references to v. Zeipel are desig- 
 nated by Z. Although duplication of material which can be found in either reference is to be 
 avoided, our experience in attempting to reproduce v. Zeipel's tables led us to fill in certain 
 gaps which are troublesome to the reader and the computer. 
 
 The first section of v. Zeipel's theory is concerned with an independent development of 
 Hansen's differential equations for ntiz and v and a repetition of the differential equation for 
 
 *t and the introduction of Bohlin's argument 6. In passing, it is well to emphasize two 
 cos t- 
 
 facts: First, the variables e and /"used throughout the theory are analogous to Hansen's e and/; 
 the dash is unnecessary, for the physically real values do not appear. Second, the constant 
 elements a, e,n,c, Q,,i are neither osculating nor mean elements; they are defined in the section 
 on constants of integration. 
 
 The perturbative function and its partial derivatives are developed in Fourier's series, in 
 which the arguments depend upon the relative positions of the disturbed and disturbing 
 bodies and in which the coefficients are infinite series in ascending powers of the eccentricities 
 and the inclination of the orbits. The coefficients in the latter are elliptic integrals depending 
 upon the ratio of the semi-major axes. 
 
 Since these elliptic integrals are functions of the ratio of the semi-major axes, or of the 
 mean daily motions, they can be .developed in Taylor's series, in which the given function and 
 its successive partial derivatives are expressed for exact commensurability and the series pro- 
 ceeds according to a small quantity w, defined by w=l 2 ft, where ft is the ratio of Jupiter's 
 
 mean motion to that of the planet and where ft differs but little from These elliptic integrals 
 
 enter the coefficients in all of the subsequent trigonometric series. Hence all the coefficients are 
 series in w. With some exceptions the terms in w, w, and v? have been used. The develop- 
 ment of all functions in powers of w is the essential principle underlying the group method of 
 determining perturbations. 
 
 The following pages contain the tables which are, in general, parallel to those of v. Zeipel. 
 At the end of sections 2, 3, 4, 5 there are brief written comparisons. To facilitate comparisons 
 
 ' Auseinandersetrung einer tweckmassigen Methode HIT Berechnung der absoluten StSrungen der kleinen Planeten. 
 
 ' Fonneln un<J Tafeln cai gruppenweisen Berecknung der allgemeinen Storungen benachbarter Planeten. Nova Acta Reg. Soc. Sc. t'psalienslx, 
 Set. Ill, Band XVII, 1S96. 
 
 Bur le Diveloppemcnt des Pertabations Flane'taires. Application aui Petites I'lanetes. Stockholm, 1902. 
 ' Angenaherte Jupiterrtcrongen fur die Hecuba-Gruppe. St. Pitersbourg, 1902. 
 
 41 
 
42 MEMOIRS NATIONAL ACADEMY OF SCIENCES. [VOLXIV. 
 
 with v. Zeipel's tables, those numerical quantities which are in disagreement are inclosed in 
 brackets. There are also certain mathematical developments useful to the reader. These 
 relations are sometimes taken from v. Zeipel and sometimes supplement his text. 
 
 Certain simple functions of the elliptic integrals y t m ' n , defined by Z 19, eqs. (73), (74), (75), 
 are tabulated in Table I (cf. Z 23). 
 
 Tables II-IVw 2 (cf. Z 26-32), giving the partial derivatives of the perturbative function, 
 are computed according to Z 24, eq. (77), by means of Table I and B 184, Tables XVI-XVIII 
 and B (Ger.) 182, Tables XII-XIV. 
 
 The elimination of Jupiter's mean anomaly from the argument gives Z 25, eq. (78), in 
 which the coefficients are derived from Table II-IV v? by the formulae given in B 61. These 
 coefficients are tabulated in Tables V-VII W? (cf. Z 33-39). 
 
 fan* i;ilal-i[ '_.-* [:! )':> i>. rM iui viii 1o p.fiu'M'i v.- Unit ,j ,z6;i iui i'.>.\ni ::. / 
 
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No. 3.] 
 
 MINOR PLANETS LEUSCHNER, GLANCY, LEVY. 
 
 43 
 
 
 
 OS.-H ^M 
 
 ioco coi> 
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 CO CM 00 O 
 
 *-CM coco 
 
 OS 0% CO CO 
 
 CO O> 
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 CM CM P 
 
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 t-co 
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 25 
 
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 53 
 
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 co *5 10 
 
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 N 
 
 
 
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 C-J t _ _ -u TJ- X' 
 
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 b- co o eo 
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 oc en .1 o 
 
 O * >! O 
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 b-!-M OS, i CM 
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 co ^J* ^ r- *o co 
 
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 CO CC O 
 
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 e e e 
 
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 CM c*. ~ C: 
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 OC O O ' CM 
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 N t^ OS 
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 CM -HOOO OS 
 
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 CM CO 
 
 CM ^H ^- I 
 
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 CO'-* 
 
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44 
 
 MEMOIRS NATIONAL ACADEMY OF SCIENCES. 
 
 [Vol. XIV. 
 
 o 
 
 Ca CO CD CM ^ COCOCO CMCOCOCO CD CO O O O O O 
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 O ^* CO OS O CD t" CD O CO CO CO i ' "^ O rH^rH^* 
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 r* OO CO CM !> CO OS CM OS i 1 O ti rH ^ CO t^- CO t* 
 
 O* O CO rH CO O* rH CO rH CM CO CO CO CO CO rH r-l r-l rH 
 
 CO 
 
 O "** i 1 O 'f OSOO QCOCDOS rH OS CM -<t< CO "tf CO 
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 o o co --. 
 
 CO O i-l CO i-HCOCOCO COCOCO rHi-li-Mj-l 
 
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 b b'o'a b 
 
 +_+_ ______ b 1 
 
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 a 
 13 
 
 B 
 
No. 3.] 
 
 MINOR PLANETS LEUSCHNER, GLANCY, LEVY. 
 
 45 
 
 
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46 
 
 MEMOIRS NATIONAL ACADEMY OF SCIENCES. 
 
 [Vol. XIV. 
 
 N T 00 I I CV| T^rH US 
 
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No. 3.] 
 
 MINOR PLANETS LEUSCHNER, GLANCY, LEVY. 
 
 47 
 
 
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 ^H CO O5 tO ^5 I s * c^ ^ 
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 ^>< r- co to to oo us 
 
 N N W CO CO ^< 
 
 e e 
 
 3 O5 
 
 co o 
 us T 
 
 o to 
 
 <* CO 
 
 
 
 a 
 
 
 
 H 
 
 O O US M CO 3 1 US -H rT O 
 
 co t- usi-i e < us g o j-j 
 t>- to t* us c^ to c>i 
 
 C5 O5 -< t~ t- O X3 US US US O 
 CO t-i-H 1C -V O CO CO OS O 
 
 ci pies i-ieo "*<< * 
 
 coco o oo to 
 o us r- t- t- 
 us to coo 
 
 Ci ^H 0t CO tD 
 O5O5 COUS t- 
 
 o rf vrim 
 
 
 e e * 
 
 t^f-HCJ'^'OO^- i IC5 Q 
 f^ T" tO US Cl * 
 
 ^cotooousc^ c ~ to 
 ^> e^)< t-f-i IM v c^ t~ 
 
 N l~ t- COO US TO 
 
 ei o'c4 oi co m eo^< rf 
 
 us cc to 
 
 P7 ^ O5 
 USO O 
 
 r^ to to 
 
 CO CO CO 
 
 
 C5T i (CO C5 TT COCOCO CO 
 US *-H f * t^ 5* OO OO 
 Oi ' OCO CS USU5 COCCOO 
 C! CO COi-1 CO i-I -! COCO-S--^" 
 t~ l^ -V t^> t* f T i t *-< O O 
 
 r-i o c4o oj eri co *i<^<'* 
 
 cf o O to ti?c? 
 
 CCOO 00 t~ t~ 00 
 r-(i-i I-1Q O 1-1 
 
 us >o usJ&oi 10 
 
 ON IN TT TT (N 
 
 CO CO CO CO CO CO 
 
 
 rjf^ ^ t ^ i '-' >-H (-7 rH 
 
 co co oo 35 So oo 
 
 COCO 00 0000 00 
 
 oo to to to co 
 oo eoojcs eo 
 
 * b ^3 
 
 1 1 + 1 
 
 2.7 " 1. F"!. ? ^'? 
 
 iJ-+ iJ- 1 i, i +JL, 
 
 r-<^ e e s e'e's s"? 
 1 + 1 1 1 1 1 1 1 1 
 
 
 b bt*0 o *5 
 
 _ + +^ 1 1 
 
 'r-i"rt"r-l" f-("r-l"i-l>-l" ^ *? *? PT ^ ^*?'o"V 
 
 + i + i + iJL+ + ' ++JL 
 
 TT rtr-i e SRS N Rgge TT'TS esess 
 ! + ,-, 1 ^1 1 1 +1 + 1 Mill 
 
 c .. fis; C- ... sic .... sJSff- 
 
 '. '. 1 + 1 1 ++ 1 I 
 ..JL .s.s.s.e^s.e^g. 
 
 
 >'. t-lf-l * K ,O I-H t-1 r-l rti-lrlr-l 70 I-i P-H P-H i-f i-H 
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 tT <CoT iCa, oTC eCnTeC <CC (CoT-vC (CcCaTtC ftraT^flrar 
 
48 
 
 MEMOIRS NATIONAL ACADEMY OP SCIENCES. 
 
 [Vol. XIV. 
 
 O 
 
 o 
 
 CO t-- 
 
 e 
 
 Tf CM 
 CM OO 
 
 in co 
 
 CO CM 
 
 O b^ 08 
 
 CO CO O 
 
 r- 10 oo 0i 
 
 CO rH r~ CO 
 b CO rH CO 
 
 COCO rH CO 
 CO 05 CM O 
 
 t~ COIN 
 
 CMCMCM 1 CM 
 
 
 
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 OCM 
 
 rHCM 
 
 Or-icO 
 
 rHCM CM CO 
 
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 1 
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 00 CM 
 
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 OS CO O 
 
 CO CO CO 
 OS r-OO 
 CO CO CO 
 O 00 rH 
 
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 ^ CO-^f CO 
 t-CMO b- 
 
 r-i co co r- 
 
 TfO COCO 
 r-* CO CM CO 
 
 O O5p 
 CO (M O 
 CM CO C5 
 CO t^ Cft 
 CM CM in 
 
 CM COCO 
 
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 oooo 
 
 CM (?4 CM CM 
 
 
 00 
 
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 m c-5 1> 
 
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 rH 
 
 rHCM 
 
 rHCM 
 
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 r-t CO CM CO 
 
 CM CO CO 
 
 CM CM CM CM CO 
 
 
 - 
 
 1. 291778 
 
 d 
 
 CM CO 
 CM T 
 
 rHCM 
 
 n o 
 
 CM CM 
 
 CO CO r-4 
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 r-i CM CO 
 
 e 
 eo t-. co - * 
 
 O b- CM CO 
 
 CO O -^ CM 
 
 -<arH coco 
 
 rH CO CM CO 
 
 OS CO -H 
 
 ^ji in 10 
 
 rH in -H 
 CM 95 CO 
 
 CO CO CO --O t~ 
 
 CO CO CO CO CO 
 CM CM CO CM -^ 
 
 01 
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 . 
 
 1. 461775 
 
 8 cf 
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 f in 
 
 rHCN 
 
 CO O 
 
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 CMCM 
 
 00 
 b- 05 CO 
 Q CM OS 
 
 r-icM CO 
 
 "JFI? 
 
 CM 1C CO CO 
 CO t-O rH 
 
 ^ o -^ 10 
 
 CO CO I*- b- 
 
 rH CO CM CO 
 
 SOS t-- 
 as in 
 
 CM CO CO 
 
 CM COrf 
 
 CN IN CM CM 
 
 e 
 
 MS 
 
 626670 
 
 OO O 
 
 3 
 
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 CO CM 
 
 in oo 
 
 CM 00 
 rH IO 
 
 co r*- cb 
 
 CM 00 * 
 CM 00 l^ 
 
 oo -^ co 
 
 C^ S01 
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 CO O r* ^? 
 
 CM i ' 01O 
 N rH CO 0> 
 
 * CM CO 
 
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 CM CO 00 
 
 CM CMC4 C<1 CO 
 
 3 
 
 
 rH 
 
 rH CM 
 
 CMCM 
 
 rH CM CO 
 
 rH CO CM CO 
 
 CM rfcO 
 
 CM CMCMCM CO 
 
 ^ 
 
 
 
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 rH 
 
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 5 in 
 
 f-CO 
 
 ss 
 
 os r-- 
 
 ^ CO 
 
 00 CM 
 
 sss 
 
 CO CM CO 
 
 o co m 
 
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 gb- CO CM 
 
 lO Tf* CO 
 O i-H rH 
 CO O CO 
 
 05 0S O CO 
 
 m 
 
 CO COrH 
 CO t" t^> 
 
 0500 Tf 
 
 t^ in co 
 
 CMCM CO 
 
 in in in >n e 
 
 OOOO rHOO 
 
 rH 
 
 
 rH 
 
 rHCM 
 
 CM IN 
 
 CMCM CO 
 
 rH CM COCO 
 
 CM CO CO 
 
 (N CMCM IN COCU 
 
 ^ 
 
 
 
 CO 
 
 i 
 
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 s|E 
 
 I-H CM 
 
 olos 
 
 m 31 
 
 CO t~ C 
 
 ^ C5 CO 
 
 O5 o e 
 CM' CM co 
 
 e e 
 
 C 7 1 OC O 
 CO 00 rH la 
 
 ScMO 
 CO-* rHCO 
 
 CMCM CO CO 
 
 00 00 CM 
 in CM rr 
 
 O) --I CM 
 CMO 00 
 
 oirfco 
 
 O5 05 O5 O5 
 00000000 -Wf- 
 
 oooo coco co in 
 CM CMCM'CM' t- ** 
 
 m t^ co 
 
 CO CO CO 
 
 M 
 
 CM 
 
 rH 
 
 
 
 in co 
 
 CM OS 
 
 CM CO 
 i i b- 
 
 SCO 
 m 
 
 l^CO 
 
 CM os 
 
 CO Oi CO 
 CO OO rH 
 
 e e e e 
 
 CO rH O 01 
 
 CO ^f CO CM 
 CD t- rH b- 
 
 t~ CO CM 
 
 e e 
 
 CO CO CO CO 01 
 OO GO OO OO t*~ 
 CO CO CO OO *O 
 
 t-fco* 
 
 SSE: 
 
 CO 
 
 
 CM 
 
 CMCM 
 
 rHCM 
 
 CMCM CO 
 
 CMCM COCO 
 
 CM CM CO 
 
 CM CM CM CM CO 
 
 > 
 
 - 
 
 1. 770420 
 
 *- CM 
 t- CM 
 
 <Nin 
 
 CMCM 
 
 C35 CO 
 CM CO 
 
 CMCM 
 
 CO CO rr 
 
 m o eft 
 
 CM COIN 
 
 e e e e 
 
 t- 00 CO CO 
 CO - CO CM 
 05 CO 00 CO 
 O ^ b--^ 
 b- CM CO rH 
 
 CM* CO CO CO 
 
 CM CO rH 
 
 r- CO CM 
 
 ip S (M 
 ^ 
 
 CM CO CO 
 
 c e 
 
 CO CO CO CO CM? CO 
 
 TJ. ^. -^. -r< co CM 
 
 b- b- b- b- 03 -^ 
 
 SCO CO CO CO b- 
 0S 05 01 OO CO 
 
 CM CM CM CM CO CO* 
 
 e 
 Sf 4 in 
 CM b- 
 
 OS rH p 
 
 ^ in co 
 
 t~ O 00 
 
 1 CO * rf 
 
 o 
 
 rH 
 OS 
 00 
 
 CO 
 CO 
 
 e s 
 
 CO OO 
 CO CO 
 iO iO 
 CMCM 
 
 rH rH 
 
 rH 
 
 CO CO 
 CO 00 
 CMCM 
 
 C>4 CM CM 
 
 10 10 in 
 
 ifl iO iO W3 
 
 CO CO COCO 
 
 CO CO CO CO 
 
 O5 in O5 
 
 rH CO rH 
 
 CM t~CM 
 
 rH CO rH 
 
 CMCM CM CM S fi 
 
 05 05 05 O O 
 S iO u3 0S 01 
 rH t rH rH CO CO ^ 
 CO CO CO CO rH rH t 1 
 
 IS 
 
 t* 
 
 
 rH 
 
 CMCM 
 
 CM IN 
 
 CMCMCM 
 
 CMCMCM CM 
 
 CMCMCM 
 
 CM CM CM CM CO CO* 
 
 SCO 
 
 e 
 
 r-i 
 
 1 1 
 
 +7 
 
 "? T 
 1 r-,1 
 
 i t'r- ('r-l"rH' 
 
 + + 1 1 
 S S S S 
 
 1 1 1 1 
 
 CM" cT 
 
 j^Pj^-* g* r- . p^gj^g* gj 
 
 1 I 1 1 III + 1 1 1 1 
 
 b 
 
 cTcT +"?*F 
 
 ^Tff +7? +7f 
 
 
 e 
 
 1 
 
 -4- 1 
 
 i i 
 
 CM ?CM 
 4- ' 1 
 
 4-14-1 
 
 1 1 1 
 
 ,-JrHrHr^ l-!l 'cO ?CM?Ci 1-5 1- 
 
 + |4-| +11 'I'l 1 H 
 
 , r- rH r. ? ? f 1 T *" 
 
 - 1 + 1 '.'.'. '. '. 1 
 
 
 s 
 
 s s 
 
 s s 
 
 
 S fi S S 
 
 s s s 
 
 ** gjjjjj* j-a-jjjjj* ?Ji 
 
 
 
 o 
 
 3 
 
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 ^o^ 
 
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 K*tt cWc? Vc,- 
 
 
No. 3.] 
 
 MINOR PLANETS LEUSCHNER, GLANCY, LEVY. 
 
 49 
 
 00 
 
 eo 
 
 S 
 
 00 
 
 CO 
 
 * 
 
 ss 
 
 <N <M U5 
 
 O O :o 
 
 ^i "T CO 
 
 CO CO 
 
 (N M d 
 
 CO CO CO 
 COCO CO 
 
 
 
 
 
 
 110379 22- 
 
50 
 
 MEMOIRS NATIONAL ACADEMY OF SCIENCES. 
 
 [Vol. XIV. 
 
 Unlt- 
 
 rH ^ rH CO CO 
 
 t* CM O CO CM 
 
 tO I s *- OO tO CO 
 
 CO CO OS lO rH 
 
 rH r" CM 
 
 CN CO 
 
 i-TrH rH r 
 
 rH rH CO 
 
 CO 
 
 tOO CO _ 
 f rH OS r-l 
 O OS 
 
 e e e 
 
 CN C55 OO 
 
 CM CO CMCM^ 
 
 1C CO CO CO 
 CM CO rH 00 
 t~ Utl t^> O 
 
 _ _ . tO tC iO tO 
 
 O rH CO CM CM CM CM 
 i I rH ^ OO OO 
 
 CO CO CO CO 
 
 
 CM CM CM CM 
 
 OO CO OO OO 
 
 CO T -f CO CO CO CO 
 
 ets e 
 
 * CN 
 
 CM CO CM CO CM CO 
 
 OS bO 00 
 
 1*5 rH Oi 
 
 Oi ^ 
 
 S8S SSS; 
 
 > t~ CO CN CNCN ! 
 
 > rH 5> CN CN CN CN 
 
 CO CO CO CO 
 
 4. 49 
 
 Oi CO OS iO 
 
 SS 55SSS 
 
 r-i cocoir 
 
 CO CO CO CO 
 
 oi CN co 
 
 1 
 
 CO CO CO CO 
 
 CO OS 00 
 
 CN t~ t- 
 - lOrH 
 
 CN OS OS OS C 
 t-1>< Tl< " 
 
 CN CO ^t* CO CO CO CO 
 
 M 
 
 ij 
 
 
 
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 CO rH CO t^ CO 
 
 in o -J* co CN 
 
 * t^ co oo m 
 
 ClOr 
 t- rH C 
 Oi rH CO 
 
 O O'O 
 
 - 
 
 1C to tO lO 
 
 to to to to 
 
 oi CN co r-i co coeoco eococ^'i COCN-^ co co eo co 
 
 c e 
 
 CO 
 
 b- CO 
 
 s II l 
 
 O OOt^- COO tOC^ICN ^ff^p-^ 1 
 
 O CMOO tOCD OCMO5 OCOCOO 
 
 to CCCM iO*tf* CMt r*- tOCOOSCO 
 
 CM CM co CM co coeoco coeoco^ 
 
 o coc 
 
 O5 rH C 
 
 COCO CM 
 CO CO ^ 
 
 CO CO CO CO 
 CO CO CO CO 
 
 CO 
 ^ 
 
 ^ 
 CN 
 
 CM CO 
 CO 00 
 O CM 
 
 _e e e 
 
 O CO O 
 C) CO !> 
 
 CM CO COCOCO CO CO CO ' 
 
 CO CO C CO 
 
 CO CO CO CO 
 
 2 g 
 
 s s 
 
 e e 
 
 VJ" CN CN e 
 
 r~ b- CN CN oo f 
 
 eo co co co rH ( 
 
 < I-H 00 OO rH ( 
 
 OJ CN CN CN' 
 
 . . os ^'* 5"* m us 
 
 SO-^O CTSOSOSOS bt^ 
 
 O OS oz -^ ^f ^ ^)* i~H rH 
 
 -_)CN * (N * CNIMCNCN COCO 
 
 I-H TT I-H co co co co m cc m co co co co co 
 
 CO CO CO CO CO CO CO CO CO CO CO CO CO CO 
 
 m 10 
 
 CMCN 
 
 m us 
 
 c 
 
No. 3.] 
 
 MINOR PLANETS-LEUSCHNER, CLANCY, LEVY. 
 
 51 
 
 vj O7 CO 
 
 S Sg 
 
 fli'V 
 
 S3 
 
 -I.- e essescc 
 I + I I I I I I I I 
 
 '. '. I +11 ++ I I 
 
 J^R^ ,,e.s s s.e.c. 
 
 C c C O 
 
 5 
 
52 
 
 MEMOIRS NATIONAL ACADEMY OF SCIENCES. 
 
 [Vol. XIV. 
 
 3 
 
 a 
 
 s 
 
 
 
 s 
 
 
 
 
 
 8 
 
 
 
 
 
 IN 
 
 
 
 
 
 
 e B 
 
 e 
 
 
 
 00 
 
 t-. i*ri 
 
 Ol^CO 
 
 
 
 CO 
 
 i t t* 
 
 
 
 o 
 
 
 Oi CM 
 I> Oi 
 
 
 
 
 IN 
 
 c4 co 
 
 coco 
 
 
 
 
 e e 
 
 O CM 
 
 Ol OS 
 
 
 
 00 
 
 O OS 
 
 
 
 00 
 
 a> 
 
 CO 00 
 
 cq t* 
 
 
 
 CO 
 
 efi co 
 
 I-HOO 
 
 
 
 IN 
 
 <NCO 
 
 eoco 
 
 
 
 55 
 
 ?,? 
 
 df s 
 
 
 t~ 
 
 
 M CO 
 
 ^J 
 
 
 
 * 
 
 t-CO 
 
 on 
 
 
 
 <N 
 
 c<i co" 
 
 coco 
 
 
 
 
 K e 
 
 
 
 
 01 
 
 i 
 
 rH rH 
 ^ CO 
 
 oS 
 
 1 
 
 
 s 
 
 COCO 
 O9 CO 
 
 rH lO 
 O 00 
 
 rH 
 
 to 
 
 
 <M 
 
 W CO 
 
 COCO 
 
 4) 
 
 
 
 e m s 
 
 
 b- ^ 
 
 
 CO 
 
 
 f< CO 
 
 
 IB 
 
 s 
 
 OS CO 
 
 3S 
 
 0> 
 
 
 e4 
 
 IM'CO 
 
 c<ico 
 
 ^ ^) 
 
 
 
 B B 
 
 
 
 
 
 CO 
 
 CO (M 
 
 
 3 e 
 
 * 
 
 s 
 
 s 
 
 il 
 
 i i 
 
 OO b 
 
 
 IN 
 
 CO CO 
 
 <N' co 
 
 ^ 
 
 CO 
 
 00 
 
 f'-O 
 !M 
 OO 
 
 O CO 
 
 i 1 i 
 
 
 
 rHlO 
 
 
 o co o 
 
 
 IN 
 
 COCO 
 
 e-i co 
 
 ^ ^ ^ 
 
 
 
 B B 
 
 
 c 
 
 
 
 1 
 
 i 
 
 00 rH 
 W Q 
 
 CN tO 
 
 00 CO 
 
 CO 
 
 lO rH O* rH 
 CM CO lO CO 
 
 SSI 8 2 
 
 (M rH C4 O 
 
 
 94 
 
 CO CO 
 
 coco 
 
 V-<ti -* 10 
 
 
 
 6 B 
 
 
 B B 8^ 8 
 
 - 
 
 * 
 
 i-H 
 
 8 
 
 I! 
 
 00 IO 
 
 Z3 
 
 00 rH 
 
 fx. CO CO CO CO CO OO C"q C"l 
 
 CO ^4* O O O O i~ t O O 
 
 
 IN 
 
 COCO 
 
 CO CO 
 
 ^ ^i^^i^ uj Otd 
 
 
 
 B B 
 
 
 B Tf 
 
 o 
 
 1 
 
 00 00 
 
 CO CO 
 CO CO 
 
 to 10 
 
 ( t rH 
 
 Tf T 1 irt iC W5 lO 
 CO CO lO "0 lO lO 
 O O CM CM CM CM 
 
 
 IN 
 
 CO CO 
 
 COCO 
 
 CO CO -^ ^ V 2L 
 
 
 
 
 
 + ?+? 7 7 
 
 c 
 
 
 
 
 rH rH rH rH rH ^H rH . ^ i CM 
 ,,+M ,,,,+M+ +J,+ + ' + + rL, 
 
 
 
 s s 
 1 1 
 
 rH r t 
 + 1 
 
 S SSS N IN SSSS rHrHrHS SSSSS 
 r^l III +r-,l 1 1 1 1 +1 + 1 Mill 
 
 c*. ... cfffi .... c?sJS* 
 
 
 l 
 
 + 1 
 
 1 1 
 
 '. 1 1 + 1 '.'.'. +1 + 1 '. '. '. 1 + 1 + 1 1 
 
 
 
 
 > g ...?H 
 
 !! 
 
 
 
 3 
 
 C?C? 
 
 O 
 
 ^-^^ ooo oooo - -. -. ooooo 
 
 
 i 
 
NO. 3.] MINOR PLANETS LEUSCHNER, GLANCY, LEVY. 
 
 Logarithmic. TABLE IV. 
 
 53 
 
 Unlt-l". 
 
 
 n 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 a 
 
 
 -Ro-o 
 
 ni-n-1]-^ 
 
 1. 898161 n 
 1. 898161 
 
 2. 283141 B 
 2. 283141 
 
 2. 153770 B 
 2. 153770 
 
 2.006171 B 
 2. 006171 
 
 1. 847875 B 
 L 847875 
 
 1.68250 B 
 1.68250 
 
 1. 51210 B 
 L 51210 
 
 
 U 
 
 n+1. -n+1 
 
 +K 7 
 
 2. 522558 
 
 2. 683079 
 
 2.528880 
 
 2. 366191 
 
 2. 197675 
 
 2.02490 
 
 1. 84887 
 
 
 ^j 
 
 
 
 2. 522558 
 
 2. 937464 
 
 2. 958044 
 
 2. 924776 
 
 2.858077 
 
 2.76886 
 
 2. 66352 
 
 
 *!'o 
 
 n+l!-n-l 
 
 W 7 
 
 2. 522558 n 
 
 2. 683079 B 
 
 2. 528880 B 
 
 2. 366191 B 
 
 2. 197675 B 
 
 2. 02490 B 
 
 1. 84887 B 
 
 
 
 n 1. n 1' 
 
 -^ 
 
 2. 522558 B 
 
 2. 937464 n 
 
 2. 958044 n 
 
 2. 924776 B 
 
 2. 858077 B 
 
 2. 76886 B 
 
 2. 66352, 
 
 
 ^0-1 
 
 n. n+2]+j! 
 
 / 
 
 2. 812563 B 
 
 3. 024413 B 
 
 2. 794447 B 
 
 2. 523495 B 
 
 2.197675, 
 
 1. 76164 B 
 
 0. 74650 n 
 
 
 Rtt-t 
 #0-1 
 
 n.-nl+K 7 
 n. nj if 
 
 
 2. 522558 B 
 2. 522558 
 
 3. 024413 B 
 2. 462558 
 
 3. 07659,8 n 
 1. 724281 
 
 3. 058902 n 
 1. 856833 B 
 
 3. 001314 n 
 2. 093957 n 
 
 i 12968 B 
 
 2.81684 n 
 2. 09513 n 
 
 
 RO-I 
 
 n.-n-2]- 
 
 y 
 
 2. 812563 
 
 3. 261391 
 
 3. 246209 
 
 3. 190606 
 
 3. 108845 
 
 3.00884 
 
 2. 89541 
 
 
 RI-O 
 
 n . _+!]-(-,, 
 
 y 
 
 
 3. 49405 B 
 
 
 
 
 
 
 
 RI-O 
 
 n 2. n+1 
 
 +^ 
 
 
 3. 36728,, 
 
 
 1- V c :' < 
 
 
 
 
 
 
 
 
 
 r t-t ^ 
 
 
 
 
 
 
 RI-I 
 
 n 1. n+2' 
 
 +T / 
 
 
 
 3. 70912 
 
 
 
 
 
 
 RI-I 
 
 n+1. nl+7 
 
 j 
 
 3. 30370 
 
 
 
 ^r * 
 
 
 
 
 
 R\-i 
 
 n-L-n+jr' 
 
 3. 30370 
 
 
 
 
 
 
 
 
 RI-I 
 
 n+1. n a 
 
 f 
 
 3. 3037,0,, 
 
 
 
 
 
 
 
 
 RI-I 
 
 n 1. nj a 
 
 j 
 
 3. 30370 B 
 
 
 
 
 
 
 
 
 R-i 
 
 n.-n+ll+jr 7 
 
 
 3. 81842 B 
 
 
 
 
 
 
 
 RO-I 
 
 n.-n+lj-ii 
 
 j 
 
 
 3. 21895 
 
 
 
 
 
 
 
 RO-O 
 
 "n 1. n] o 
 
 +T / 
 
 2. 72638 B 
 
 
 
 
 
 
 
 
 RO-O 
 
 n+1. n]+c 
 
 +7^ 
 
 2. 72638 
 
 
 
 
 
 
 
 
 
 a 1. n+2 
 
 5+^ 
 
 
 
 2. 94112 
 
 
 
 
 
 
 RO-O 
 
 +!. nl+o 
 
 *f 
 
 2. 72638 
 
 
 
 
 
 
 
 
 RO-O 
 
 71 1. n] < 
 
 ~* 
 
 2. 72638 B 
 
 
 
 
 
 
 
 
 RO-O 
 
 n. 71+1]+?! 
 
 / 
 
 2. 51524 
 
 2. 84832 
 
 2. 78195 
 
 2. 69286 
 
 2. 58759 
 
 2. 47019 
 
 
 
 RO-O 
 
 n. n 1] r 
 
 j 
 
 2. 51524 B 
 
 2. 84832 B 
 
 2. 78195, 
 
 2. 69286 B 
 
 2. 58759 B 
 
 2. 47019 B 
 
 
 
 1 '0 
 
 n+l.-n+I 
 
 +^ 
 
 3. 25180 B 
 
 3. 45486 B 
 
 3. 34235 n 
 
 3. 21906 B 
 
 3. 08741, 
 
 2. 94903 B 
 
 
 
 75 
 
 Xv^ *0 
 
 n 1. n+1 
 
 -hS 
 
 i 25180 B 
 
 3. 62946 B 
 
 3. 66471 n 
 
 3. 66409 B 
 
 3. 63529 B 
 
 3. 58453, 
 
 
 
 R 
 
 n+l.-n-l 
 
 -^ 
 
 3. 25180 
 
 3. 45468 
 
 3. 34235 
 
 3. 21906 
 
 3. 08741 
 
 2.94903 
 
 
 
 RI-O 
 
 n-l.-n-i; 
 
 -tf 
 
 3. 25180 
 
 3. 62946 
 
 3. 66471 
 
 3. 66409 
 
 3. 63529 
 
 3. 58453 
 
 
 
 RO-I 
 
 n. n+2]+a 
 
 j 
 
 3. 49076 
 
 3. 69598 
 
 3. 53278 
 
 3. 33224 
 
 3. 08741 
 
 2.77380 
 
 
 
 RO-I 
 
 n. n]+^' 
 
 
 3.25180 
 
 3. 69598 
 
 3. 76576 
 
 3. 78484 
 
 3. 76831 
 
 3. 72575 
 
 
 
 RO-I 
 
 n.n]n / 
 
 
 3. 25180 B 
 
 3. 33141 B 
 
 2. 99523 B 
 
 2. 24789 B 
 
 2. 51136 
 
 2. 76863 
 
 
 
 RO-I 
 
 n.-n-2]n 
 
 j 
 
 3. 49076 B 
 
 3. 891 34 B 
 
 3. 91658 n 
 
 3. 90661 B 
 
 3. 87001 B 
 
 3. 81284 n 
 
 
 g 
 
 R [n.-n+l]+* 
 
 j 
 
 
 4. 36208 
 
 
 
 
 
 
 a 
 
 R [n 2. n+1 
 
 +V 
 
 
 4.18801 
 
 
 
 
 
 
 o 
 
 
 
 
 
 
 
 
 f 
 
 
 [2 
 
 RI^ 
 
 n 1. n+2 
 
 +^ 
 
 
 
 4. 52584 B 
 
 i: 
 
 ! ' * 
 
 
 
 
 fit.] 
 
 n+1. nl+r 
 
 y 
 
 4. 13780 n 
 
 
 
 
 
 
 
 
 B 
 
 n 1. nj+ji 
 
 j 
 
 4. 13780 B 
 
 
 
 
 
 
 
 
 
 n+l.-nl-jr' 
 
 4. 13780 
 
 
 
 
 
 ?y . % 
 
 
 
 j! 
 
 n 1. n' x' 
 
 4. 13780 
 
 
 5 ?t 
 
 O c^ I ft* i 
 
 
 x S? ^5 
 
 
 
 
 
 
 
 - . , ^ 
 
 '**"'' C 
 
 
 - ; br r-^ 
 
 
 
 B . 2 [n.-n+ll+* / 
 
 
 4. 60272 
 
 
 
 
 ^1 * 
 
 I 
 
 
 J? . 2 [n.-n+l]-^ 
 
 j 
 
 
 4. 07416 B 
 
 
 
 
 
 
 
 ^0-0 
 
 n 1. n] o 
 
 +X* 
 
 3. 51583 
 
 
 
 
 
 
 
 
 
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No. 3.1 
 
 MINOR PLANETS LEUSCHNER, CLANCY, LEVY. 
 
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No. 3.] 
 
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 S S,,,- ^ rt - ,_;,_ rt ' ,4 i-i 
 
 '. '. I + I I ++ I I 
 
 ss ggseess 
 
60 
 
 MEMOIRS NATIONAL ACADEMY OF SCIENCES. 
 
 [Vol. XIV. 
 
 o 
 
 M 
 
 1 1 
 
 **< 
 
 3 
 
 1 
 
 <M 
 O CO 
 
 o'oi 
 
 0000 
 I-H rH 
 rH 
 
 1 + 
 
 t-oo 
 
 rH* 
 
 co ife 
 
 * 
 
 fH 
 + 1 
 
 
 
 
 rjiTf* ^< 
 I> t^ t-^ t-^ 
 
 ooo o 
 
 00 CO 00 CO 
 
 1 I++ 
 
 
 
 
 S 
 
 ei 
 
 50 
 1 
 
 lO 
 
 CO 00 
 
 irioi 
 
 rH 00 
 CM * 
 r-H 
 
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 iriod 
 
 CO(M 
 
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 + 1 
 
 0505 
 
 O) TT 00 
 
 CD lO O 
 
 lO t~ 
 rH 
 
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 Ld O5 O5 
 
 O5 IN OS CO 
 CM O5 rH OO 
 lO t- O51N 
 
 rH O5CO * 
 
 + 11 + 
 
 m 
 gt2 
 
 CM rH rH 
 
 rt 2 
 1 + 1 
 
 OO CO CO CO 
 
 CO CO CO CO 
 
 oo oo 
 
 rH rH i 1 rH 
 
 1 1 ++ 
 
 
 OO 
 
 S 
 
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 rH rH fH tH "^ 
 
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 CNI rH 
 
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 CN rH 00 
 
 S^tn 
 
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 TiHOOCO 
 
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 SS85 
 
 rHOS^rH 
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 CO 
 O >O<N 
 
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 CO CO CO CO 
 
 ododod od p 
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 t*- t*- t^* r^ ^H 
 
 pH rHrHfH Q 
 
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 ^s 
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 lO CO -^ 
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 O ^t 1 CO ^* 
 
 *o coco 
 
 "*l IN rH Tt* 
 rHOO COIN 
 
 33 
 
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 CO 
 
 o< cot- 
 
 CO rH t-- 
 
 OO t~ 
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 rH CO 
 
 1 + 1 
 
 en o Cb c^ 
 
 rH rH |-5 fH 
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 r _( r n fH rH 
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 l l++ 
 
 1 
 
 
 
 
 CO CM 
 
 00 * 
 
 O5<N 
 
 n<(M 
 
 ^CO 
 
 IN O5CO 
 
 rHCM 
 
 00 0000 00 
 
 
 US 
 
 lO 
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 1 
 
 00 t- 
 rHO 
 IN 1> 
 CM 
 
 1 + 
 
 3S 
 
 1C DO 
 
 CO 
 
 + 1 
 
 CO lO t* 
 
 CO "^ O5 
 
 iR &t i-H 
 coco 
 
 rH 
 1 + 1 
 
 Tt^ CO CM CM 
 
 lO O CO 00 
 
 rH lfi> Tj* Oi 
 * 
 
 + 11 + 
 
 OS CM * 
 O O5 t^ 
 
 ^g^ 
 
 CO 
 
 1 + 1 
 
 CM INCMd 00 
 
 1 1 ++ + 
 
 * % 
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 o 
 
 t* 
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 1 
 
 
 CM 
 
 00 
 
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 10 rH 
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 rH lO 
 
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 + 
 
 rH 
 < 
 CM 
 
 1 
 
 COCO 
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 rH t- 
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 1 + 
 
 in co 
 
 lO rH 
 CO U5 
 
 CO 
 
 + 1 
 
 tN N CO 
 
 lO N CO 
 
 ^rnS 
 1 1 
 
 1 + 1 
 
 T-HIO<NOO 
 
 rH O5 OO O5 
 
 ** oico 
 
 rH CM rH CO 
 * 
 
 + 11 + 
 
 CO COO 
 
 <N rH CO 
 OS COO 
 
 CO 
 1 + 1 
 
 t~ t^ t~ t- TC CO 
 
 in in in in * * 
 
 rH rH rH rH CO O 
 CO CO CO CO CM 00 
 
 m 
 
 1 1 ++ 1 + 
 
 1 
 
 m 
 
 CO 
 
 o 
 
 CO 
 
 00 
 
 IN 
 
 1 
 
 CO CM 
 
 rH Tt< 
 
 n<od 
 
 *-s 
 
 CM 
 
 + + 
 
 mo 
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 od co 
 
 CO CO 
 
 CO 
 CO 
 
 + 1 
 
 U5CO 
 
 r-i c4 U5 
 rH CO" 
 t~ CN rH 
 rHIM 
 
 rH 
 
 1 1 1 
 
 oso in 
 
 ^ T}^ t^ CM 
 
 in o O5 co 
 
 OO CO O t^ 
 
 I-H rH rH *^ 
 CO 
 
 + + + + 
 
 com 
 
 l>rH CD 
 
 oo in * 
 
 IN lOCO 
 rH in 
 CM 
 
 1 + 1 
 
 o o oo 
 odododod iro 
 
 IN IN CU CM rH CO 
 
 CO CO CD CO OO O 
 CO COCO CO rH 
 IN 00 
 
 1 1 ++ M 
 
 " -_ CO ' 
 
 co feel 
 
 r^. 
 + + + 
 
 
 S 
 
 S3 
 
 00 t~ 
 
 CO iO 
 
 CO 00 CO 
 
 COIN t- 
 
 COCO 
 
 * T!<"J< * 
 
 
 W 
 
 CO 
 
 1 1 
 
 CO 
 
 1 
 
 t~ in 
 
 in in 
 
 00 S 
 ++ 
 
 >. t^ 
 
 moo 
 
 CO 00 
 
 o> 
 
 1 1 
 
 Oi OS C5 
 O COIN 
 rH CO 00 
 
 1 1 1 
 
 CO IN CM CO 
 COCO f~ 00 
 O 00 IN *< 
 
 CO-* T< Tfl 
 IN 
 
 + + + + 
 
 o co-f 
 
 00 OS IN 
 CMO * 
 IN T CO 
 rH 
 
 1 1 1 
 
 in in in in 
 
 rH rH rH rH 
 
 OOOO 
 *** -^ 
 
 1 1 ++ 
 
 ^ CO O 
 t~ rH O 
 1C CO O 
 CO ^<'CM 
 (N COCO 
 
 + + + 
 
 
 
 
 t*- lO 
 I> iC 
 
 CO-* 
 CM O5 
 
 <N O5O5 
 
 CNJ t-CO 
 
 OO 
 
 in in in in 
 
 
 FM 
 
 S 
 
 <M 
 
 1 
 
 N lO 
 
 3 
 ++ 
 
 OOr-i 
 OO CO 
 
 oo t~ 
 
 rH 
 1 1 
 
 rH O5 t- 
 <Mt-Q 
 00 t~O5 
 rH>Tl< 
 
 1 1 1 
 
 CO C<J t- 5 
 
 CO^ COt-H 
 Si 1 ! 1 rH 
 i> r- ^ 
 
 rH 
 
 ++++ 
 
 rH OS CO 
 (NO rH 
 rH CD -^ 
 TP t~O 
 
 1 1 1 
 
 SO O O rH OJ 
 CD CO CO OO CO 
 CM INCMCM COO 
 
 **** 00 i-H 
 
 i i ++ 7J 
 
 CM m co 
 
 2 g? S 
 
 00 * CO 
 
 co in ^ 
 1 1 + 
 
 
 
 
 CO CD 
 
 e-i <N 
 
 88 
 
 O rHO 
 
 COrHrHCO 
 
 l~CO t~ 
 
 CMCMNIN 
 
 
 O 
 
 l- 
 
 <N 
 
 r-t 
 
 1 
 
 r*- r- 
 t- r- 
 
 lOiO 
 
 ++ 
 
 ifl O 
 gg 
 
 1 1 
 
 CO CO CO 
 1OO 1C 
 lO rH 1C 
 rH CO rH 
 
 1 1 1 
 
 CO CO OO CO 
 rH O O rH 
 CO 35 O5 CD 
 * CO CO-O* 
 
 + + + + 
 
 CO ^ CC 
 
 1 1 1 
 
 CO CO CO CO 00 OO 
 rH rH rH rH in in 
 rH rH II rH t^ t 
 
 CMCMCMCM og 
 
 1 1 ++ ++ 
 
 in m 
 
 OS OS 
 
 ss 
 
 coco 
 
 ++ 
 
 
 
 
 "5"? 
 i i 
 
 rH~rH~ 
 
 n 
 
 "s~ 1? 
 1 ^1 
 
 rH T-H . t r 1 
 
 ++ 1 1 
 s s e e 
 1 1 1 1 
 
 <N~ C? 
 
 +^,1 
 g 5* 5* 
 
 b b "ts ^o 
 
 +JLiJL ~. _ 
 
 rH i 1 r-! rH rH rH 
 
 +11+ _~^ ^+^ 1 
 gees ses 1-1 S--H g 
 III! Ill + 1 1 1 
 
 b 
 
 C? STcM* J? ? S" 
 
 +^^ i i ^^^ i ' i 
 
 esssg r V r r c ? ff ? 
 Mill +M + 1 1 
 
 j ^ pft . 
 
 
 1 
 
 1-HrH 
 + | 
 
 1 1 
 
 IN | N 
 + ' 1 
 
 rH rH rH T 1 
 + 1 + 1 
 
 i i i 
 
 + 1 + 1 +11 ' 1 '. 1 
 
 7+7^7 111 1.17 
 
 
 
 
 o 
 
 -"?:. 
 
 o o 
 
 vE^S- 
 
 ^s,, 
 
 O 
 
 ^-S-S-S- 
 
 v~5~5. 
 
 ^-S-S^ -EvS.^- .SxS-S-S 
 
 OOOO OOO .-.^.M 
 
 ~5S s ,.. S-S-^- -vE^. 
 
 ^"^^W^^N *~m~m*~m o^o^o 
 
 
 d 
 
 eye? 
 
 o o 
 
 <yo 
 
 N W C 
 
 O'G'Q' 
 
 c?c?c?c? 
 
 O 
 
 O'O'O' 
 
 ocycyc? cyo-cy O'O'O'O 1 
 
 O'O'Q'O'O' o'O'O' c^cyo 1 
 
 3 
 
 r I 
 
 > 
 
No. 3.] 
 
 MINOR PLANETS LEUSCHNER, GLANCY, LEVY. 
 
 61 
 
 
 
 m 
 
 
 
 i 
 
 rH 
 
 
 
 
 CO 
 
 
 :W. r '*.&&> i-' v 
 
 
 O <NCO 
 
 4 47 
 
 lO CO 
 
 Hi 
 
 rHIO 
 
 1 4 
 
 
 
 i-H 
 
 O30O 
 
 
 
 C$ CO 
 
 CO kO 
 
 0.B8T4' 8 .OW>1 
 
 
 4 4T 
 
 1 
 
 
 
 o 
 
 l 
 
 
 
 N S 
 4 41 
 
 74 
 
 
 
 
 CO CO OO 
 
 CO ?0 ^* 
 CO O ^* 
 O ?: iA 
 r- < r* CO 
 
 1 
 
 
 4 4 1 
 
 14 
 
 
 
 ci coo 
 GO y. " 
 
 CO CO 
 
 ->-CO r-( Q ^H 
 lO OO ^^ OO O 
 CO C5 W CD 
 CO rH O CO 
 
 1 
 
 
 4 4 1 
 
 1441 4 
 
 i S 
 
 s 
 
 H 
 
 >0 
 
 d *? o 
 
 O oo 35 
 
 < <N S 
 
 K.rtfQ < i' .(".OT S.TVfi >-'(-.-- .T- .,A 
 
 rH 00 O CO 
 CM CO CO i-H 
 I-H CO * O 
 CO 00 Q 
 
 ! '-" 
 1 4 
 
 | 
 
 4 1 1 
 
 14 14 
 
 
 
 0- -HCO 
 
 00 OO 00 
 
 1 
 
 
 S| 
 
 2 s i i i 
 
 1 
 
 
 4 1 1 
 
 44 1 1 
 
 
 
 00 
 
 
 f- t-O 
 b- f- iO 
 
 CM (NO 
 CO CO CO 
 
 1 44 
 
 
 CM OO 
 
 rHCO 
 
 4 1 1 
 
 S CM (M CO CO O 
 I-H O O ^H CO 
 CM CO KB CO S 
 
 r-4 rH O 
 
 44 44 4- 7 
 
 lO lO 
 
 CO CO 
 
 1 1 
 
 
 rH - 
 
 co r* co 
 
 CO t^ t- 
 i-HCM 
 
 + 1 1 
 
 CO -~ ^* CO OO CO 00 00 <5 Tl * J 
 
 co 222S -o 22 
 44 II 441^ 7 44 
 
 
 
 114 4 
 
 
 r-i CM CM 
 
 t- CO.CO 
 
 r1 rH -H 
 rH rH 
 
 4 1 1 
 
 coco o Q c a> <y> " 
 
 OO OO O>Ou9iO 
 COCO i If-l rHrHrHrH 
 rH rH CO CO CO CO CO CO 
 
 4-4 44 44 1 
 
 ^^7 ^ b ^47^ 
 47^ 4 T 47 ^7 
 
 rH-H g ggggg g^ 
 
 1 4 I 1 1 1 1 1 1 1 
 
 
 "e"? 
 1 1 
 
 ^-^. iZiZ I. ^ 1^, 
 
 ,_^_, ^^ 141 ^ ^ +7 7 4 "^ 1+41 4+1 
 
 rHr-H g ggg Ol CM gggg ^ r7 g gg g g 'JT 
 
 41^.1 MI4_I till 441 || 1 || 
 
 '. '. 1 4-11 44 1 1 
 
 
 i r ( ^-t 
 
 '. 4 1 
 
 xS- -S-S 
 
 O o 
 
 , CM i-H -* r-i ,,, r-l rH rH rH ? ? CM rH rH i-i i-i rH 
 
 '. '. '. 1 411 '.'.'. 4141 '. '. 1 41 4- II 
 
 ** &;&*$* 
 
 
 o* 0*0* 
 
 o o cy o-o-o. 0-00.0- o- o-o- o-o> o- oc 
 
62 
 
 MEMOIRS NATIONAL ACADEMY OF SCIENCES. 
 
 TABLE VII. 
 
 [Vol. XIV. 
 
 Unit-l". 
 
 
 n 
 
 
 
 l 
 
 2 
 
 3 
 
 4 
 
 5 
 
 e 
 
 
 fi (n _n+l)+r 
 
 / 
 
 - 79. 10 
 
 - 191. 93 
 
 - 142. 48 
 
 - 101. 43 
 
 - 70. 45 
 
 - 48. 14 
 
 - 32.52 
 
 
 fl.(n.-n-l)-7 
 
 S 
 
 + 79. 10 
 
 + 191. 93 
 
 + 142. 48 
 
 + 101. 43 
 
 + 70. 45 
 
 + 48. 14 
 
 + 32.52 
 
 
 JZ,. 'n+l.-n+l 
 
 ~\~n 
 
 + 372. 6 
 
 + 482.0 
 
 + 266.7 
 
 + 130.9 
 
 + 52.0 
 
 + 9.6 
 
 - 10.7 
 
 
 
 ~f~7t 
 
 + 293. 5 
 
 + 865. 9 
 
 + 979. 2 
 
 + 942.4 
 
 + 826. 9 
 
 + 683. 6 
 
 +542.1 
 
 
 .R.. (n+l. n 1 
 
 TT 7 
 
 - 293. 5 
 
 - 290.1 
 
 - 124.2 
 
 - 29.5 
 
 + 18.5 
 
 + 38.5 
 
 + 43.2 
 
 
 7Z,. (n-l.-n-l 
 
 -' 
 
 - 372. 6 
 
 - 1057.8 
 
 - 1121.6 
 
 - 1043.8 
 
 - 897. 4 
 
 - 73L7 
 
 -574. 6 
 
 
 Bo.,(n.-n+2)4V 
 
 - 649.5 
 
 - 1057.8 
 
 - 622. 9 
 
 - 333.8 
 
 - 157. 6 
 
 - 57.8 
 
 -5.6 
 
 
 /Z . I (n.-rt4V 
 
 
 - 333. 1 
 
 - 1057.8 
 
 - 1192.9 
 
 - 1145.3 
 
 - 1003.0 
 
 - 828. 
 
 -655. 9 
 
 
 
 
 + 333. 1 
 
 + 290.1 
 
 + 53.0 
 
 - 71.9 
 
 - 124. 1 
 
 - 134.8 
 
 -124.5 
 
 
 .Ro.j(n.-n-2)-7r / 
 
 + 649.5 
 
 + 1825.5 
 
 + 1762.8 
 
 + 1551.0 
 
 + 1284.8 
 
 + 1020.6 
 
 +786. 
 
 
 Bj. (n.-n+l)+i 
 
 r" 
 
 
 - 3119 
 
 
 
 
 
 
 
 iJ 2 . (n-2.-n+l 
 
 14V 
 
 
 - 2330 
 
 
 
 
 
 
 
 JZ,., n-l.-n+2)4V 
 
 
 
 + 5118 
 
 
 
 
 
 
 JZ,., n+l. n)+a 
 
 ^ 
 
 + 2012 
 
 
 
 
 
 
 
 
 lZ|.j n 1. n)+7i 
 
 J 
 
 + 2012 
 
 
 
 
 
 
 
 
 1Z,., n-l.-n)-^ 
 
 - 2012 
 
 
 
 
 
 
 
 
 J?J.J(TI~~I. ft) 3 
 
 S 
 
 - 2012 
 
 
 
 
 
 
 
 
 /I0.j(7l. ""Tl"^! )~7~7I 
 
 { 
 
 
 - 6583 
 
 
 
 
 
 
 
 JZ .i(n.-n+l)-K' 
 
 
 + 1656 
 
 ~r 1 
 
 IT" + 
 
 
 
 
 
 -7?o-o tt~l.~tt)""* C 
 
 '+** 
 
 - 533 
 
 
 
 
 
 
 
 
 -Ko-o *M~1- ^)~H 
 
 >4V 
 
 + 533 
 
 
 
 
 
 
 
 
 /J ., ) n-l.-n+2)-<5+7r / 
 
 
 
 + 873 
 
 
 
 
 
 
 Ro-o n+l. n)+c- V 
 .RO.O n 1. n) d n' 
 
 + 533 
 - 533 
 
 
 
 
 
 
 
 
 /Z . (n.-n+l)+r, 
 
 f 
 
 + 327.5 
 
 + 705. 2 
 
 + 605.3 
 
 + 493.0 
 
 + 386. 9 
 
 + 295. 2 
 
 
 
 Ro-o(-- n -l)-^ / 
 
 - 327. 5 
 
 - 705. 2 
 
 - 605.3 
 
 - 493.0 
 
 - 386. 9 
 
 - 295. 2 
 
 
 
 U ( n +l n+1 
 
 4V 
 
 - 1950 
 
 - 2850 
 
 - 1897 
 
 - 1163 
 
 - 643 
 
 - 299 
 
 
 
 tfl;(n-L-n+l 
 
 +* / 
 
 - 1622 
 
 - 4260 
 
 - 4923 
 
 - 5107 
 
 - 4898 
 
 - 4432 
 
 
 
 /Z,. (n+l. n 1 
 
 TT 7 
 
 + 1622 
 
 + 2145 
 
 + 1292 
 
 + 670 
 
 + 256 
 
 + 4 
 
 
 
 JR,. (n-l.-n-l 
 
 -Tt 7 
 
 + 1950 
 
 + 4966 
 
 + 5529 
 
 + 5600 
 
 + 5285 
 
 + 4727 
 
 
 
 #., n.-n+2)+^ 
 
 + 3096 
 
 + 4966 
 
 + 3410 
 
 + 2149 
 
 + 1223 
 
 + 594 
 
 
 
 RQ.I 71. tl)-|-jr / 
 
 
 + 1786 
 
 + 4966 
 
 + 5831 
 
 + 6093 
 
 + 5866 
 
 + 5318 
 
 
 
 /J 0>1 n. n) itf 
 
 
 - 1786 
 
 - 2145 
 
 - 989 
 
 - 177 
 
 + 325 
 
 + 587 
 
 
 
 RQ.I ft. ft 2) 7 
 
 / 
 
 - 3096 
 
 - 7786 
 
 - 8252 
 
 - 8065 
 
 - 7413 
 
 - 6499 
 
 
 8 
 
 fl 3 . (n.-n+l)+j 
 
 [ 
 
 
 +23018 
 
 
 
 
 
 
 | 
 
 JZ 3 . (n-2.-n+l 
 
 14V 
 
 
 +15418 
 
 
 
 
 
 
 (2 
 
 RL n-l.-n+2)4V 
 
 
 
 -33562 
 
 
 
 
 
 
 JZ,. n+l.-n +7 
 
 !* 
 
 -13734 
 
 
 
 
 
 
 
 
 JZ-i. n 1. n +7 
 
 t* 
 
 -13734 
 
 
 
 
 
 
 
 
 JZi. n+l. n 7 
 
 r 7 
 
 +13734 
 
 
 
 
 
 
 
 
 JZ,. n 1. n 7 
 
 / 
 
 +13734 
 
 
 
 
 
 
 
 
 .Ro.2(n. n+l)+7 
 
 I 
 
 
 +40061 
 
 
 
 
 
 feS 
 
 
 Rf,.^(n. n+l) 7 
 
 I 
 
 
 -11862 
 
 
 
 
 
 
 
 "0*0(1 1. n) e 
 
 r+TC 7 
 
 + 3280 
 
 
 + f 
 
 
 
 
 
 
 /Zo.o(n+l. n)+i 
 
 I+;t/ 
 
 - 3280 
 
 
 
 
 
 
 
 
 R t . a (n 1. n+2 
 
 
 
 
 - 5854 
 
 
 
 
 
 
 JZ . (n+l. TI)+ 
 
 r It' 
 
 - 3280 
 
 
 
 ^ i-2 ""* 
 
 
 
 
 
 Ro-o( n ~l- n) < 
 
 t-V 
 
 + 3280 
 
 
 
 ta *"* ^ 
 
 
 
 
 
 R . (n. n+l)+7 
 
 r 7 
 
 
 - 1303 
 
 
 - 1127 
 
 
 
 
 
 /Z . (n.-n-l)-j 
 
 r* 
 
 
 + 1303 
 
 
 + 1127 
 
 
 + 838 
 
 
 
 B,. n+l. n+l 
 
 4V 
 
 
 
 
 
 
 
 
 < 
 
 fi,! n-l!-n+l 
 
 
 
 
 
 +13600 
 
 
 
 
 P 
 
 /f,. n+l.-n-l 
 
 -TT 7 
 
 
 - 7080 
 
 
 
 
 -14465 
 
 
 2 
 
 ^,.0 n-1. n 1 
 
 -T/ 
 
 
 -12290 
 
 
 
 
 
 
 1 
 
 ,Ro.].(n.-n+2)+; 
 
 ? 
 
 - 7475 
 
 
 
 
 
 
 
 
 *^0 ' 1 x ~~ ^/ "l ^* 
 
 
 
 
 -14430 
 
 
 
 
 
 
 I\Q.^\Tl. ~~ft) ^~7t 
 
 
 
 
 + 4532 
 
 
 
 
 
 
 R^n.-n^)-, 
 
 C 7 
 
 + 7475 
 
 
 
 
 +20370 
 
 
 
NO. 8.] MINOR PLANETS LEUSCHNER, GLANCY, LEVY. 63 
 
 With these tables we compute terms of the first order in the mass in Hansen's differential 
 equations for the function W and the perturbation in the third coordinate. See Z 7, eq. (33) 
 and Z 8, eq. (39). The first order parts of the equations are expressed in Z 41, eqs. (82), (83), 
 in the form of trigonometric series, hi which the coefficients are computed from the formulae 
 given hi B 67. These coefficients comprise Tables Vlll-XIVto 2 (cf. Z, 42-48). 
 
 Table XV (cf. Z 50, eq. (88)) is an auxiliary table of the same type of construction, which 
 is employed in the computation of terms of the second order hi the mass hi the differential 
 equation for W (cf. Z 53). 
 
 * 
 
 " 
 
 
 - 
 
64 
 
 MEMOIRS NATIONAL ACADEMY OF SCIENCES. 
 
 [Vol. XIV. 
 
 ci m * co 
 
 ss *" 
 
 rH O 
 OOCO 
 
 8 
 
 CM t- OS 
 
 _ CM in co os 
 
 CO rH in CO t-* 
 r-b- rHCMCO 
 
 e 
 
 ii 
 
 OS TT CO lO OOOt-- 
 rHOOlOb- COb-iO 
 CMCOOCM COrHb- 
 - 
 
 t* CM b- CM 
 
 Tt*COTfCO 
 
 IH rHOJ CMCM CM'roco 
 
 b-m COrH 
 
 CM coco** 
 
 CO CO OS 
 
 CM COCO 
 
 CMC<ICMCM 
 
 e 
 
 t-- CO CO ^T 
 
 CO t^ CM b 
 
 rH Tf-^ 1 COCO 
 
 o coos Ito 
 
 rH CO 1 ^ OCO OrHiO 
 
 S 
 
 e e 
 
 F-H CM CM 
 OSO OS 
 
 STf 
 
 CM CMCO COCO CO - 
 
 88 8 
 
 b- coin CMCO os CM in 
 
 - Q rH CMOS OCMOS 
 
 CJ5 lO CO CO in 1 s ** rH 
 
 oo coco in-* CM i f co 
 
 CO CR b- CMCO CM CO b- 
 
 rH i-icJ WC<i CMCOCO 
 
 SCOO&OO CO rH Tt 
 
 OS CO lO CM rH CO 
 
 CO CD rH O CO CM rH 
 
 t OO ** O CM CO **P 
 
 t-- in * CM b- r- o 
 
 CM co -3< CM co -^ 
 
 COrHCO rH 
 
 Ti CO Ut> O 
 ^* CO ^ CO 
 
 CM CM' CM CM 
 
 CM CM* CO* 
 
 OCMCOO 
 COffCM. 
 
 CO***^ 
 
 s e 
 
 iOCJb- 
 
 o^t>- 
 
 C-1CMCO 
 
 COtr-OS 
 iC^CO 
 
 
 
 CO COb? * 
 
 CM CM -^ 
 
 2 CM ** 
 ^ in 
 in o oo co os 
 
 rH 
 
 CM (N CM CM 
 
 SOS CM CO 
 CO CO CM 
 
 CD t- CO t CM 
 CM COCO 
 
 CM CO CO 
 
 b- t- co ^F 
 
 OS t OS CM 
 CM CM CO m 
 
 - 
 
 -g*io^ 
 
 OS CO CO 
 CM f*- CO 
 CO OS CM 
 I- t-- r 1 
 
 i co t- co 
 
 8S 
 
 OS COCM 
 COOS 
 
 iftCO 
 
 CM CO* CO *** 
 
 CM CO T 
 
 CM CM CM C i 
 
 1 -- 
 
 cp 1-1 ci toco 
 
 CO 00 5O t-H t^- 
 
 ci 
 
 ^ 
 
 CD 
 
 CMCO roco 
 
 O r-t CO 
 
 COCOOS ^-HC 
 COCOCO iCOiC 
 TCMO lOO 
 
 co -^ * 10 
 
 <N 
 
 Tf* 
 CO 
 CO 
 
 co 
 r 
 
 8 8 
 
 O OO t-- 
 
 COCM m OS 
 
 ^ 00 Oi CO 
 
 CO^t 1 in b- 
 
 coco cooo 
 
 i i os Tf o 
 
 8 
 !>. in r- 1 
 
 -^OrH 
 
 O CM OS 
 
 - 
 
 S S 
 
 O:r- I O OS 
 CM CO CO CO 
 CO OS CO CO 
 CO *n CM CO 
 
 -j< t*- m co 
 co co 10 co 
 
 88 
 
 CO OS CO OS 
 CM CM CM CM 
 
 OOCOOOCO 
 
 r- ici^noi 
 
 STO 
 rHCM 
 
 rH t^-CO 
 OS ^fCM 
 
 iCCO 
 
 COOS 
 iC 
 
 COCOlO b-r-iO^ 4 
 
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No. 3.] 
 
 MINOR PLANETS LEUSCHNER, GLANCY, LEVY. 
 
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 C4COM OOOOOO r* ,* *4 r+ r*, r* ,+ 
 
 o o o *-< - ~ * 
 
 S" 7 1 ? ? ! 
 
 
 
 
 
 
 
 
 
 
 
No. 8.] 
 
 MINOR PLANETS LEUSCHNER, GLANCY, LEVY. 
 
 71 
 
 
 >'yoi'-* 'ze'n'x 'i x'x 
 
 : :. -W 
 ^D OD 
 
 & 
 
 -3 5 '2 
 
 f-H 
 
 i 
 
 ' l-l 
 
 
 + 
 
 >n.--x -.-- -i V^ 
 
 + + + 
 
 t=5 4 i 
 
 '-"-- ~^T. i- 
 
 1 + 
 
 * } Z-f. "H- | , 
 
 O QO 
 
 o o -C -* n cs 
 
 T OO Ci t~-* O C^l 
 
 TT i++ 7 
 ji i MJI e 
 
 4 ;f 
 
 i-H ^" 
 OS t^. i-H OS 5? 
 
 l 
 
 ^ rH CO i-H 
 
 II 11 + 
 
 i^ 
 
 eccc^" Oil < u5?sco 
 
 C-JOCO WN C^iCO 
 
 voces ceo rtco 
 
 t^-t^-CO OOO COCiCt 
 ^ ^H CO N COCO 
 
 +4- 1 1+ 1 + 1 
 J H H Jl H M H 
 
 o cs oc oo 
 
 ++7 + 
 ji n ii n 
 
 r < -H [, f^ _ 
 
 - - * C"- ~ ~- 
 i-H F t i-H i-H 
 
 +1 1+1+ 
 Jl II II II II II 
 
 oo"o o 
 
 rH p i rH rH 
 
 T++ T 
 
 i i 
 
 * 7 ? +7 
 
 w to '"" v *o *s ^"^ b b *"<& *"- ^ *Q 
 
 ^^ iJL7JL+ +'7+7+7 
 
 6eo<to 7 ^7^ 
 ++7^ +F+^7 
 
 VT" 1 ? 77 s 7"r 8 sssssss 
 + KS tiiiti IMI . " '. 
 
 777 ggssg 
 tit 1 - IIIM 
 
 '.'.'. '. '. 1 ' ' 1 +11 ++ I | 
 
 1 1 1 1 -L I _L 1 I 
 
 
 _8^g^g^ g g g g e 
 
 
 - - 00000 
 
 ^ 
 
 , w , 
 
 ' -i -: - : 
 : TZ X X 
 
72 
 
 MEMOIRS NATIONAL ACADEMY OF SCIENCES. 
 
 [Vol. XIV. 
 
 B 
 
 s 
 
 = 
 H 
 
 
 rH rH rH rH 
 
 od od od od 
 
 CM IN IN CM 
 
 
 H| 
 
 i ++ 1 
 
 + 11 + 
 
 
 
 11" VT TP 
 
 
 
 
 OOOO 
 
 OO CO OO CO 
 CO CO CO CO 
 
 
 
 1 ++ 1 
 
 + 11 + 
 
 
 
 rH I-H rH rH 
 OOOO 
 
 ssss 
 
 CM 
 
 rt 
 
 1 ++ 1 
 
 + 11 + 
 
 rH 
 1 
 
 
 C4 CN NN 
 
 rH rH rH rH 
 
 So o o 
 CO CO CO 
 
 
 
 1 ++ 1 
 
 + 11 + 
 
 
 
 rH rH rH r-i 
 i 1 rH rH rH 
 
 OOOO 
 
 1 
 
 
 1 ++ 1 
 
 + 11 + 
 
 
 
 rH rH rH rH 
 
 <M O4 CM C<* 
 
 CO CO CO CO 
 
 
 O 
 
 
 + 11 + 
 
 
 P 
 
 ++ * F 
 
 ++f f 
 
 V N 
 
 ti t: 
 
 + 1 
 
 
 ++ 1 1 
 g g g g 
 1 1 1 1 
 
 ++ 1 1 
 
 g g g g 
 
 1 1 1 1 
 
 + 1 
 s 
 1 1 
 
 
 + 1 + 1 
 
 OOOO 
 
 + 1 + 1 
 
 1 1 
 
 
 fc,fc,*,fc, 
 
 ft.*.*,*, 
 
 K 
 
 
 
 
 
 
 
 9 
 
 B 
 < 
 
 H 
 
 rH rH OOOCDrH OO O OO CO 
 
 OOOO t-^ CO Oi CD t>^ OO 
 10 CM t~ 
 
 - . CO CM 
 OOrHO 
 
 + 1 
 
 00 00 
 
 I I ++ +++1 
 
 + 1 
 
 ^t COO t^ 
 
 OJ CO C^ CO t-^ CO M^ ^ 
 
 rH O rH rH O rH C 
 rH rH rH f 
 
 I I ++ +++ 
 
 *i 
 
 i 
 
 i- 
 
 US 
 
 U5 
 
 + 1 I I ++ +++ 
 
 S 
 
 OD 00 
 
 ^" rf CM CO t~ rH CSC5OOO 
 
 + 1 I I ++ ++ I I 
 
 ire lOi 
 o o CM oo 
 
 + I 
 
 I ++ ++ I I 
 
 i* *n \G 
 
 it i! i; n 
 
 f^ 
 
 8 It 
 
 JOS rHO"<NOS CicOCOOS 
 
 5 LOCOES- ss 
 
 + 1 I I ++ ++ I I 
 
 
 IT II 
 
 :- -- '.- 
 
 
 
 V V V 
 
 K ts 
 
 ^^ ++ I I 
 
 rHr-, g g g g 
 
 + 1 I I I I 
 
 i? 
 
 V S + + 
 
 K ti w 
 
 +^ \J^ 
 
 g 'g"g"g"g~ rH rH g g 
 I I I I I ++ I I 
 
 . '. + I + 
 
 L I I + I + I '. '. I + 
 
 ~*~o*~c **~^~~~~~Z?~~ ^~^~^> o o 
 o o 
 
 K'l"i. 
 
No. 3.] 
 
 MINOR PLANETS LEUSCHNER, CLANCY, LEVY. 
 
 73 
 
 
 iO *& 
 
 
 rrcc t~ e 
 
 
 
 
 OCCNg 
 
 ""S^S 
 
 
 
 i + 
 
 ++I 1 
 
 111 + 
 
 + i 
 
 
 ee 
 
 -- -- 
 cc cc 
 
 cc cc 
 
 ^ 
 
 C-J OC CC^ 
 ^H 1(5 ! 
 
 
 
 1 + 
 
 + +I 1 
 
 111 + 
 
 
 
 CO 
 
 cc cc 
 
 O C5 
 
 i-H SO rH r^- 
 
 C^ CC I s " lO 
 
 
 
 1 + 
 
 ++ 1 1 
 
 
 
 
 cc cc 
 
 K3US 
 S3 
 
 O CC C5 'V 
 
 0-HON 
 
 
 
 
 C^l S f-H |-i 
 
 to ^ oo 
 
 * 
 
 ! 
 
 1 + 
 
 + + 1 1 
 
 1 I++ 
 
 
 
 
 
 1 
 
 
 
 " " 
 
 CC5C C^ t^ 
 
 ^ S ^* oc o 
 
 Cl Ci ^ t- 
 
 I 
 
 
 1 + 
 
 + + 1 1 
 
 1 I++ 1 
 
 + 
 
 
 
 
 
 
 O) 
 
 1C U3 
 
 cc cc 
 
 r- o 3; cc 
 
 C^4 ?O * O 
 
 <CtOCOCO lOtO O O 
 CICCGOO ^^H *** ^** 
 
 S 
 
 
 1 + 
 
 
 1 1 ++ +1 1 1 
 
 l 
 
 V 
 
 
 
 
 
 \\ 
 
 
 \\\\ 
 
 -> s . ^ +. v 
 b b K b< b k 
 
 \ 
 
 i i i 
 
 \\ 
 
 ^X^-C^^J^ 
 
 \ \ V + +v ^V ^ V V ++ 1 I 1 
 
 1 v 
 -*~- fc 
 
 els"? 
 
 iJL 
 
 
 
 ++ 1 1 
 
 e e e e 
 i i i i 
 
 \\jL +^ s;++^j^ +j^ J^+NJ^ 
 
 7 j^ 
 
 1 1 1 
 
 g e 
 
 
 
 ^c"e g +1 1 1 1 1 1 ++ Mill 
 
 J; g 
 
 l-f ! 
 see 
 
 -S-S- 
 
 + 1 + 1 
 
 -~5~5-S 
 
 .m* 8 - e g-g-gte ii gtetff 
 
 ii 
 
 
 o o 
 
 - _ _ - 
 
 c c c i co ~* r* * r* v+ nn c c c c c 
 
 - 
 
 o c o 
 
 CMCNJ 
 
 C5C5C3C3 
 
 SSgg tte> oootocs to e o cteotfe. 
 
 M CM 
 
 
 
 
 01 JO JJK J 
 
 t -o^ 
 
 
74 
 
 MEMOIRS NATIONAL ACADEMY OF SCIENCES. 
 
 [Vol. XIV. 
 
 
 rH rH 
 
 O t- OS 00 
 
 CO O CO CO 
 
 
 coco 
 
 
 
 
 
 33 
 
 * r-ilrfOS 
 
 10 co o t~- 
 
 SCM CO<M 
 
 
 OS OS 
 
 *C CO O CO 
 COCN 00 W 
 
 Ci rH CO OS 
 
 
 
 
 rHf~ rH t~ 
 
 CO I-" O 
 
 
 
 rH t- OO CD 
 
 
 
 HI 
 
 
 
 rH 
 
 
 
 rH ^ 25 
 
 lO CO 
 
 
 
 1 + 
 
 + + 1 1 
 
 111 + 
 
 
 + 1 
 
 1 I++ 
 
 +++I 
 
 
 
 m id 
 
 * OS 00 
 
 t-ocooo 
 
 
 OS OS 
 
 
 ; ^> 
 
 
 
 .. 
 
 coosSco 
 
 ESOCOGO 
 
 
 coco 
 
 00 CO 
 
 SCO rH t^ 
 
 ColoCMrH 
 
 
 * 
 
 
 CO GO rH OS 
 
 1-IOrHCO 
 
 
 coco 
 
 
 COCO CO^JI 
 rHO t^. 
 
 
 
 1 + 
 
 + + 1 1 
 
 111 + 
 
 
 + 1 
 
 1 I++ 
 
 + + + 1 
 
 
 
 CO CO 
 
 CO 00 00 CO 
 
 COCOOSO 
 
 
 oo 
 
 
 
 
 
 rH rH 
 
 OO 
 
 rH rH 
 
 10 co coin 
 
 *f O CO rH 
 
 OO "3 rH rH 
 
 CO -f t- lO 
 CO rH 1C 
 
 
 So? 
 
 38 
 
 OV CO t~^ lO 
 
 
 CO 
 
 
 
 
 
 
 O CO 
 
 CM CO CO 
 
 
 
 1 + 
 
 ++ 1 1 
 
 111 + 
 
 
 + 1 
 
 1 1 ++ 
 
 ++ 1 1 
 
 
 
 33 
 
 CM* CM 
 
 rH rH 
 
 CM CO t^i-H 
 
 ^ rH rH -^ 
 O5 CM "t> CD 
 CO rH lO CM 
 
 o oso oo 
 coco coco 
 
 CO OS 1O CO 
 CO rH t^o 
 
 rH t- 
 t~ CO 
 
 00 ^ 
 
 CO CO 
 
 CO Ci OO *** 
 I- 1C rH rH 
 
 O rH OS CO 
 
 rH CO OO lO 
 Tf 00 OS CO 
 
 m 
 
 <N 
 
 
 
 
 CO 
 
 CO CO 
 
 CO O CO CO 
 
 COIO CO 
 
 CO 
 
 
 1 + 
 
 ++ 1 1 
 
 1 I+ + 
 
 + + 
 
 + 1 
 
 1 1 ++ 
 
 + + 1 1 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 CO CO 
 OS OS 
 
 00 CO OS 00 
 
 OO 00 CO CO 
 
 
 coco 
 
 
 
 
 
 rH rH 
 OS OS 
 rH rH 
 
 f- t- K3 OS 
 
 1/3 U3 CO $" 
 
 oo coco 
 
 t^ t-o 10 co 
 
 IO lO OS CO OS 
 O O CO 00 CO 
 
 CO 00 
 
 1C lO 
 
 CD rH rH 
 CO CO t^- 
 
 CD CD lO CO O 
 
 CD CD ^f OO CO 
 
 
 
 1 + 
 
 ++ 1 1 
 
 1 1 ++ 1 
 
 1 + 
 
 + 1 
 
 1 I++ 
 
 ++I 1 + 
 
 
 O 
 
 OO 
 
 rH rH 
 
 oioi 
 1 + 
 
 OS CO 00 t- 
 OS CO O rH 
 
 o i^ com 
 ++I 1 
 
 OS CO CO OS 
 SCOCOT(I 
 CO CO CO 
 
 1 I++ 
 
 rH rH CO CO 
 + 1 + + 
 
 coco 
 
 CO CO 
 
 + 1 
 
 CO OS * t~ 
 CO *9* O t~* 
 OS OS CO CO 
 CO rH CO CO 
 
 1 1 + + 
 
 CO CD CD CD 
 
 O5 CO CO 0i 
 
 CO rH rH CO 
 
 ++I 1 
 
 U5 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 1 . 
 
 %J 
 
 
 
 
 
 " 
 
 1 
 
 i+li 
 
 i-H rH rH rH 
 
 ++ 1 1 
 
 + > 4 
 
 T^v V l| -C*Jj 
 
 -K V v v + + 7 M 
 
 !Ji++ 1 f + 1 T +^+7 
 
 i> 
 
 rH rH r-H rH 
 
 ++ 1 1 
 
 Y >-| 
 
 c^^'VcT i+ 
 
 
 
 + 1 
 g g 
 
 1 1 1 1 
 
 -)-+ \ "\ rH g 
 
 g'g^'g'g + 1 
 
 ggggg rHr-H ggggS 
 1 1 1 1 1 ++ 1 1 1 1 1 
 
 + 1 
 
 g g 
 
 1 1. 1. 1. 
 
 + + 1 | rH g 
 
 g glTg + 1 
 
 c s 
 1 I 
 
 
 1 1 
 
 + 1 + 1 
 g g g s 
 
 i i i i n* 
 
 II rH* II rH rH , , rH rH rH rH rH 
 
 1 + 1 + 1 '. '. 1 + 1 + 1 
 
 1 1 
 
 g g 
 
 + 1 + 1 
 
 g-g-i-g- t"' 
 
 rH i t 
 
 1 + 
 
 
 o o 
 
 O O O O 
 
 rH r- -* v~5' 
 
 ^^S'>S"5^S- ^S^5- "S^S-v i~ *~- 
 
 o o 
 
 o o v ~o o 
 
 ^~*~*~*~ ^S'^S 
 
 ^H'^S' 
 
 
 
 
 
 ^-i-t^^-^ OOOOO 
 
 
 
 
 -< 
 
 
 CO CO 
 
 CM CM CM CM 
 
 faq tc; fc) ID K5V? 
 co coco co WH^H 
 
 tq'ijjtq'^tq' kf* 3 ? bf'D'i^flD'lD 5 
 
 tejtej 
 
 tej tej fej tej 
 
 CM CN CN C^ 
 
 tqlDtDU) K?b? 
 
 CO CO CO CO ^ "N 
 
 hi* 
 
 
 
 
 
 
 
 
 m, jo 
 
 wa. 
 
 w 
 H) 
 ta 
 
No. 3.] 
 
 MINOR PLANETS LEUSCHNER, GLANCY. LEVY. 
 
 75 
 
 
 . 
 
 
 7 
 
 
 *S ?: : si 
 
 
 I 
 
 "V 
 
 -. 
 
 
 
 
 * S 
 
 3k 
 
 *^ *^1 J. -1 -'I 1 
 
 -1 ^> -"? * -^ 
 
 
 C-J C^ 
 
 H 
 
 CO CO 
 
 J5 
 
 u 
 
 ^i 1 "T" 
 
 ^* >5 39 P9 ^ , 
 
 -, 
 
 ss 
 
 M 00 
 
 CO C5 
 
 + 1 
 
 K : lr 7 
 
 
 
 + 77 
 
 < 
 g 
 i 
 
 N V ~T~V s 
 
 
 vvv - - ++ 1 1 1 
 
 febb tits o ;q- to <a 
 
 "sT'sTI? Ztt "g^'e's^s" 
 
 III ++ 1 1 1 1 1 
 
 V V ++ 1 1 V 
 
 +1 ?^77 jkji 
 
 r-<-H gggS <^+|^ 
 
 +i i i i i +~ i 
 
 S g ^- S g g S 
 
 7+1 1. 1. 7+ 1 + 1 
 
 SSSSS 
 
 e J. + 1 + 1 '. '. '. '. 
 
 oc oooo ^ 
 
 L-'.--,-* i.?h? ^fcPhfcPh^k^PK-? 
 
 55 ^" ^S^ 
 
 
 t n JOJDBJ 
 
 " J 
 
 ^ - 
 
 ^1 
 
 i 
 
 " II X -I 
 
 S? 1 
 
 ++ i i 
 
 s 
 
76 
 
 MEMOIRS NATIONAL ACADEMY OF SCIENCES. 
 
 [Vol. XIV 
 
 s r 
 
 CO CO 
 
 
 CO ^^ 
 
 O Oi 
 
 -I 
 
 t~ 
 
 CO OS t~ 
 
 O O5 Tf< 
 
 CM 00 CO 
 
 CO 
 
 + + 
 
 com 
 
 CO CO CO 
 
 i 1 1 i r~ 
 
 r-lin CO 
 CN 
 
 1 + + 
 
 CO CJ O 'O t-* CO 
 
 CO C^l CO *O *0* O5 1 
 
 f-i *^ c; ic co o 
 
 + + 7 
 
 CO 
 OS COO4 
 
 t^ O CO 
 
 co i i u5 
 
 1 1 ^ 
 
 + + 
 
 t- a> co 
 
 Ci CO CM SNI 
 
 CM N CO 35 
 
 ii co 55 1-1 
 
 + 1 + 
 
 "tf 
 o" ct r-. CN co -( 
 
 CO CO tft> C<J O OS 
 
 C$ b* OO 
 
 SOS OS 
 IMO 
 
 * rH iO 
 
 t~O 
 
 OS CO C 
 rH |- Ci 
 
 OO 
 
 r^ t^- co !> I s - t 
 
 OS i ( lO * 1 ^* r- 
 
 ^s ^a g 
 
 s ..i- 
 
 3 
 
 n rHlS 
 
 + + 
 
 CO QO OS 
 
 + + 1 
 
 cc eocN ?i t^ oo 
 
 rH ^ CM CO CO O 
 CM COCO W CO ft 
 
 + + 1 
 
 iO W CO 
 
 + + 
 
 t* CO 
 
 CO O f""" CO OS 
 
 + 1 + 1 
 
 C<I 
 
 C^CO^'CCO ^PTO O 
 
 + + rt +7 + 
 
 fH 
 t* t- CO 
 
 ifS os co 
 
 + i 
 
 0r-l C~. 
 
 ^i-l 3lO CO 
 + + II 1 
 
 CO 
 
 os r^ ?o r^ o *^ c-i 
 
 ^tf CO CO CO CO O Cr 
 N rHCO CO 
 
 o 
 
 
 1 
 
 e e"? 
 
 ! ' i-i r-i 
 . + 
 
 i 11 IrHrH i-H rt i H rH 
 
 rn^? "s? "5" sess e? cT ssss 
 
 t- '.-?'. '. '. '. '. t^-e '. - '. '- 
 
 7 CM , CXI i-i I-H r-l r-i , , i-H rH i r- i 
 '..+'. 1 +1 + 1 . '. '. + + 1 
 
 MM OOO NC*C>. OOC 
 
 rH r- 1 r-( rH 
 ^^^ ^.^ ^ ^ ++ II ^ ^ 
 
 ^ + i-i TIM +^ 
 
 fi SSS -C- .... ccc 
 
 rHrH (MTC-l f-HrHrHl-1 
 . + '. . + '. 1 + \ + I . '. 
 
 05 OQ'CQ' 
 
 oo ci c-t ~t M *HMM oeo occo 
 
 O i~ OO CMCMN __. OO 
 
 
 
 XI JOpB J 
 
 1 
 
 M 
 
 s 
 
No. 3.] 
 
 
 
 
 SO -^ -^ n 
 t- O t~ 
 * 1-1 o 
 
 00 
 
 r ~ 
 
 S) rH V O CO 
 
 O cc r- ^ 
 
 99 O 
 
 
 OS 10 O * l~- 
 
 OC O I s * O ^' 
 
 jg 
 
 O O ^^ O 'O 
 
 8 
 
 w l~^ o o^ 
 W CM 
 
 + 
 
 + + + 
 
 
 
 
 to cot- asm 
 r~ co i-i T 
 
 ^1 !-! 
 1 ( I-H 
 
 
 
 ; f? S 4 
 
 
 
 
 
 + "+7 
 
 
 + i + 
 
 s s s s 
 I 1 1 
 
 9^ J^*" 
 
 x*s. 1 
 
 s rt'rt' R a 
 
 + + 1 
 
 s s s s 
 
 s s e 8 a 
 
 a o o o 
 
 o o o 
 
 o r o o a 
 
 tfcf coV 
 
 
 : P JOWBJ 
 
 MINOR PLANETS LEUSCHNER, GLANCY, LEVY. 
 
 13 If. 1 . ,;-..-LViM HIT TO fO 1 -".^O-TVii 
 
 77 
 
 
 
 fll ori) 
 
 
 ffi 'Ki'*^ ^d4 lo 
 
 
 
 
78 MEMOIRS NATIONAL ACADEMY OF SCIENCES. 
 
 INTEGRATION OF THE DIFFERENTIAL EQUATION FOR W. 
 
 With the exception of Tables LVI and LVII all the following tables are concerned with 
 the integration of functions whose coefficients can be derived, more or less directly, from the- 
 preceding tables. The terms of first order in the mass, before and after integration, are of the 
 
 type , A 
 
 where . C v . q = C . p .g + C V p. q -w + C^p.g-w 2 H ---- (see Z 25) 
 
 and A = [n + r-%(n-s)]s + (n-s)6+i IL+i' II' 
 
 In the argument A the factor n is always a positive integer; the factors r, s, i, and i' are 
 
 Tc 
 positive and negative integers. Evidently, the factor of is -~ where k is any positive integer, 
 
 and the arguments in a series are I n I r I s A. Within the extent of Bohlin's tables all of the 
 coefficients can be written in symbolic form from B 188, XVII, XVIII. In the notation for 
 the coefficients the particular values of r and s are given, and there remains to be found only 
 
 the positive value of n, if there is one, for each multiple of -~- 
 
 || 
 
 The following tables present, in skeleton form, any series of the given type. There are 
 properly two tables, one for perturbations in the plane of the orbit, and the other for perturba- 
 tions perpendicular to the same. The headings A and I are defined by 
 
 J = n-n' 
 
 Considering first the tables referring to the plane of the orbit, omitting for the moment 
 the arguments bearing the subscripts 5 or <r, the argument A for any term is read from a 
 
 Ice 
 main heading -5- and the first two columns under this heading. The tabulated numbers are 
 
 the respective factors of 0, A, and I. The degree of the factors in the eccentricities is indicated 
 in the subscriptsy-g'in the symbol for the coefficient. Further, when j tt = 1 
 
 Hence the coefficient of A is also the number n in the proper table of the numerical values of 
 the coefficients. For instance, in the function T 2 (Z 41, eq. 82) we have for one term 
 
 where F, taken from Table VIII, is numerically 
 
 F,. (n-l. -)n_4=- 1514" + 5780"u>-8976'V. 
 
 Adding e </> to the argument and taking the coefficients from Table IX, we have also in the 
 function T, -- 
 
 n-l.-n)^ sn 2e- 
 where G V9 (n- !.-)_= +452"- 1475"w+ 1451"^. 
 
 In this manner the series is built up. 
 
 The coefficients having subscripts d and a belong to terms depending upon the mutual 
 inclination of the orbit planes. They differ from the preceding type of terms in three ways. 
 In the first place the subscript signifies the addition of J and 2" to the argument, respec- 
 tively. Evidently, if J is added to the argument, the factor of A is not n but nl, from 
 which we determine n. Lastly, these terms contain the factor f, i. e., within the extent of 
 our tables the exponent t is not greater than unity. 
 
 For the tables referring to functions which concern the perturbations in the third coordi- 
 nate the same explanations hold, with the exception that the additional subscript rc f signifies 
 the addition of 11' to the argument. 
 
 These tables, in connection with the proper tables of numerical coefficients, enable the 
 computer to write a complete series by inspection or segregate any term of given degree and 
 given argument. 
 
No. 3.] 
 
 MINOR PLANETS LEUSCHNER, GLANCY, LEVY. 
 
 79 
 
 
 H 
 
 
 r-T 
 
 - 
 
 -- " ~ <O oo 
 
 
 
 
 :- 1OO ICO 
 
 
 N 
 
 
 | 
 
 ! 
 
 
 
 * 
 
 
 
 h 
 
 "7 
 
 fc 
 
 * 
 
 n^ -r= _ oo^ o=^ 
 
 
 * 
 
 B ^t- ^ ^^00 , _00 
 
 
 5 
 
 7 
 
 1 
 
 " 
 
 ^ o o 
 
 
 ? 
 
 - - - 
 
 
 a 
 
 "7 
 
 t 
 
 * 
 
 CO CH-, CO^ OV<0 0<0 
 
 
 * 
 
 1 1 1 
 
 
 N 
 
 7 
 
 "I 
 
 i 
 
 MO ecoc* -H IN 
 1 
 
 
 2 
 
 -.-. m m -. 
 
 . 
 
 2 
 
 7 
 
 "T" 
 
 - 
 
 
 
 * 
 
 
 
 * 
 
 
 4 
 
 " 
 
 <O f "" r- 
 
 
 
 
 e "cc 
 
 
 * 
 
 7 7 
 
 1 
 
 -> 
 
 7, _ _ 
 
 
 
 
 O M OT M M v M 
 
 
 N 
 
 "7 "77 "77 
 
 c? 
 
 a 
 
 n co o-roe on,- * _r-,, CHCO nt .a ^cs = ^=o n^. . N^^W^MW 
 
 
 
 
 C4= ^C O^X 0,,* 0X> OX,, NOO N 00 NNOCO r.00 <NOC0 NOO^eNO 
 
 
 H - 
 
 i 7 77 
 
 r 
 
 
 
 .O M o ^ o, ^ eo o r* ^ ec o o ^ 
 1 
 
 
 
 
 o o w N o-* o , o * * * o o-^t o 
 
 
 ^ 
 
 7 "77 "77 
 
 + 
 
 -i 
 
 C4 O ^> v-ieo r.- eOi^iC OC4V *-":7i7. O^<OC t-OOCOt^ 74 ^-7J-^ :7 ^ r: : - ^- ^ O * CO * CO ~H C 1 * 
 
 
 * 
 
 7i O, O, 7) ' 717]' 7 i 71 "C OC4C4O,OOO,^>OCOO,,CCO,3OO,X,O, O,XO,O^< 
 
 
 ''< 
 
 7 77 77 
 
 
 | 
 
 0^ ^ W K<* *** 
 
 
 
 
 o N cs o^ o^ o-* * o w c*ei<oe*eci<o wo MtDC4M M M ri M 
 
 1 j 
 
 
 1 
 
 ~~^ IZ ^ ^,^. ^. **** * +^ 
 
 J9 S-J -5- -s- ++H {? s- +n?e"S-e =?=! -S^il CCS? Islsc? ||p 
 
 I-J: B !=! *""" fcfi tttt tititlt ttttt ill Iltlit ttttttt 
 
 
 
 '. + i '.'.+'. i +i+7 '. i. >. +7+7 +7 i Lj '.7 7+7+7 '. '. '. '. '.7 '. 1 7 +77++77 
 
 
 
 5 55. -55. 5-5-5. 5555 555 55S5 55 5555 55555 555 5\5\5555 e e e e a e e 
 
 
 
 
80 
 
 MEMOIRS NATIONAL ACADEMY OF SCIENCES. 
 
 [Vol. XIV. 
 
 it 
 
 + 1 
 
 c 
 a 
 
 E 
 
 m 
 
 2 + 
 
 s 
 
 o* 
 
 h 
 
 
 7 
 
 r 
 
 O O 04 
 
 r* O 
 
 _U 
 
 7 
 
 CO M 
 
 N 
 
 
 7 ~7 
 
 T -, 
 
 <M -f o -t- c-t -J r^eoeoua ci --o o -* c 
 
 
 
 
 
 
 
 i i 
 
 1 
 
 n 
 
 
 rH I-H 
 
 1 1 
 
 i 
 
 TJ M M rj- 
 
 -* *H co o on < co o 
 
 . 
 
 eo ec ^o 
 
 rt " >0 --""" *..5 
 
 N 
 
 
 7 7 
 
 + 
 
 CSO^ -HCC OV NO 
 
 - CO CO-,10 ONO(N- M U5NCN 
 
 
 
 1 1 
 
 1 II 1 
 
 - ,__, . *< 
 
 
 7 
 
 _ 
 
 7 "* 
 
 
 o o 
 
 * 
 
 w 
 
 (N <N 
 
 N 
 
 
 ^-. -. 
 
 
 1 
 
 
 cS ^ 
 
 
 
 
 
 040 0^00 0^-^00 WWWO 
 
 1 1 "*"j 
 
 = 
 
 
 ' 77 
 
 r - 
 
 ^ o ** eo 
 
 O (N O <N 
 
 
 
 N O) -* 
 
 O O *J> ^ O * 
 
 I 
 
 
 7 "7 
 
 + ^ 
 
 
 ; 
 
 . 
 
 O -V NMtO OO*WtO Of * OO 
 1 
 
 OOT,0-,^.00 0-.<=-00 00 OOO^T, 
 
 ; 
 
 
 7 7 
 
 - 
 
 
 
 
 
 P4 ~ ~f o o * ri d<o 
 
 CM <N O)C>4^0 M d<O C4 C4 ~~ r > 7 1 
 
 I 1 
 
 Itz : 
 
 tev +ili \ -t ivilvl 
 +^ ++77 ivvA +i+7JL7 
 
 2 aeaa f? , i ST s 'aa'e 
 + 1 1 1 1 1 frSc-l 1 + 1 1 ' 1 
 
 e a 'S^B''?'? a a S?QS^S' e'a'Ja 9'aa'? 
 1 1 1 1 1 1 1 1 + + ^ + e ^ 1 1 1 1 MM 
 
 
 \\ +7+7 "Ti +T c i'+f c i 1 
 
 +7+7+7+7 iTi.Tii +7+7 +7+7 
 
 
 ^^ O^O^O ^i^^Ti^H '^O^o'^^^'*^ 
 
 .^-5-5.5.-S. 5-5-S.S5-5. 5-S-S-S. S.5^. 
 
 
 
 
 -I 
 
NO. s.] MINOR PLANETS LEUSCHNER, CLANCY, LEVY. 81 
 
 Our problem is now the integration of the partial differential equations Z 7, eq. (33), Z 8, 
 eqs. (37) and (39), and Z 9, eq. (47 1 ). 
 
 In the trigonometric series to be integrated the argument is a function of 6, s, <f>, A, I. 
 The last two are constants. According to the principles of Hansen, <f> occurs outside the opera- 
 tion. Numerically, however, it is equal to s. The argument 6 contains e implicitly. See Z 9, 
 
 eq. (43). Hence we must, in general, write 
 
 F(e, d) 
 and 
 
 = _ 
 ds ds 50 ds 
 
 In order to set up the partial differential equations from the total derivative, the following 
 notation is introduced: 
 
 F(t, 0)=[F(s, ff)] + F(t, 0)-[F(s, 6)] 
 
 where [F(e, 0)] signifies that part of the function which is independent of e. Again, since s has 
 the period of the planet, there can be no secular terms in e (with the exception of the function 0), 
 i. e., 
 
 
 On the other hand, the argument varies much more slowly, and there may be secular terms 
 in 0. Hence 
 
 and may occur outside the sign of integration. 
 
 Owing to the presence of the required function in the differential equation, the integrations 
 must be performed rank by rank where rank is defined as follows: 
 
 In the course of the developments there arise negative powers of w. Since w is a small 
 quantity, these factors increase the numerical value of the terms, or, in other words, they lower 
 the order. Therefore, it is better to define order in terms of both the disturbing mass m' and w. 
 For this purpose v. Zeipel makes the assumption that both w and -^m' are quantities of the 
 first order. Order so defined is called " rank," and the word "order" is reserved as usual for 
 
 7/1 ' a 
 
 the powers of m'. The factors 5- are arranged according to rank in Z 53. 
 Any function is then written in the form 
 
 where the subscript denotes the term of lowest rank, for F { (s, 0) contains terms of more than 
 one rank since each coefficient is itself a Taylor's series in w. In assigning rank it is to be noted 
 that the coefficients in all the preceding tables contain the factor m' implicitly. The implicit 
 mass factor is indicated at the foot of each table which follows. 
 
 On the basis of the foregoing principles, the differential equation for W, 
 
 d_W == dW + bW d0 = T 
 ds ds d0 ds 
 
 expressed in Z 52, eq. (91), is broken up into four equations, Z 53, eqs. (95, 95 4 ), according 
 to rank, and before integration they are again subdivided according to parts which contain e 
 and parts which are independent of s. The total derivative is then in the form of eight equivalent 
 equations, and the integration can be performed in the following order: 
 
 W t ; F 2 -[FJ; [FJ; F 8 -[FJ; etc. 
 
 It is possible to avoid the computation of T 3 , as v. Zeipel did, by the introduction of some 
 auxiliary functions, but we found it preferable to tabulate them. 
 
 Employing Table XVo, and by inspection of Tables VIII, IX, X, XI, T t is written directly. 
 (Thas no terms of first rank.) 
 110379 22 - 6 
 
82 
 
 MEMOIRS NATIONAL ACADEMY OF SCIENCES. 
 
 [Vol. XIV. 
 
 TABLE XVc. 
 
 r. 
 
 TJnlt-l" 
 
 
 Sin 
 
 w 
 
 w 
 
 1C* 
 
 
 -J 
 
 - f+# 
 
 + 43.1 
 - 43.1 
 
 - 128.0 
 + 128.0 
 
 + 171 
 - 171 
 
 
 + 20+2J 
 2t-<j>+28+24 
 <l>+28+24 
 
 + 271. 5 
 - 67.1 
 - 294.9 
 
 - 636.6 
 + 108. 2 
 + 740. 6 
 
 + 526 
 + 32 
 - 734 
 
 
 2t+ 48+44 
 3t-<l>+48+44 
 t+<l>+40+4J 
 
 + 159.9 
 - 45.7 
 - 167.4 
 
 - 593.3 
 + 122. 1 
 + 637.4 
 
 + 869 
 - 174 
 - 984 
 
 
 2i+<{>+68+64 
 
 - 81.7 
 
 + 418. 9 
 
 - 907 
 
 9 
 
 28+24 
 i-<l>+28+24 
 - t+<l>+28+24 
 
 -1180 
 + 273 
 +1496 
 
 +2962 
 - 179 
 -4265 
 
 - 2935 
 - 1170 
 + 5572 
 
 9 
 
 2t-<!> 
 9 
 
 - 173 
 
 - 211 
 + 384 
 
 + 512 
 + 899 
 -1410 
 
 - 684 
 - 1921 
 + 2605 
 
 n 
 
 e+ 48+44 
 2 t -<l>+48+44 
 f+48+44 
 
 -1514 
 + 452 
 +1679 
 
 +5780 
 -1475 
 -6656 
 
 - 8976 
 + 1451 
 +11172 
 
 n 
 
 2t+ 28+24 
 St-<l>+28+24 
 c+<j>+20+24 
 
 - 6 
 - 83 
 + 136 
 
 + 408 
 + 262 
 - 878 
 
 - 1307 
 - 564 
 + 2285 
 
 ? 
 
 2i+ 60+64 
 3c-<l>+68+64 
 ,+<i,+68+64 
 
 -1149 
 + 360 
 +1227 
 
 +5902 
 -1734 
 -6415 
 
 -12820 
 + 3301 
 +14400 
 
 ^ 
 
 2 t +</>+48+44 
 
 - 102 
 
 + 112 
 
 
 i 
 
 2t+<l>+&8+84 
 
 + 750 
 
 -4900 
 
 
 5' 
 
 28+ 4 
 t-<!<+26+ A 
 - c +<l>+28+ 4 
 
 + 318 
 + 222 
 - 646 
 
 -1081 
 -1012 
 
 +2452 
 
 + 1552 
 + 2227 
 - 4296 
 
 1 
 
 t+ ^ 
 2t-<{>+ A 
 <i>+ * 
 
 + 130 
 + 112 
 - 285 
 
 - 484 
 - 565 
 +1211 
 
 + 808 
 + 1393 
 - 2475 
 
 n' 
 
 t+ 48+3J 
 2e-<l>+48+3J 
 $+48+34 
 
 +2279 
 - 580 
 -2460 
 
 -7160 
 +1410 
 +8138 
 
 + 8896 
 - 520 
 -11342 
 
 n' 
 
 2t+ 28+34 
 3t-<!>+28+34 
 t+<f>+28+34 
 
 - 314 
 + 127 
 + 291 
 
 + 702 
 - 399 
 - 537 
 
 - 90 
 
 + 598 
 - 478 
 
 Y 
 
 2t+ 68+54 
 3t- </>+68+54 
 t+<!>+68+54 
 
 +1887 
 - 542 
 -1974 
 
 -8417 
 +2221 
 +9002 
 
 +15550 
 - 3377 
 -17350 
 
 I 
 i 
 
 2e+<{'+48+54 
 
 - .".'," >..,;..". stirp') 
 Zt+j+W+U 
 
 + 390 
 -1263 
 
 -1556 
 +7397 
 
 
 9 
 
 .-# 
 
 - t+<!> 
 
 + 568 
 - 568 
 
 - 3106 
 + 3106 
 
 
 * 
 
 48+44 
 ,-<f>+46+44 
 - i+<I>+48+44 
 
 +6716 
 -2114 
 -7960 
 
 - 26627 
 + 6488 
 + 33462 
 
 + 44700 
 
 P 
 
 t+ 28+24 
 2t-<f>+28+24 
 <{>+28+24 
 
 /, 4- 128 
 + 535 
 - 978 
 
 - 3166 
 - 2505 
 + 7431 
 
 - 23105 
 
 
 
 m' 
 
No. 8.] 
 
 MINOR PLANETS LEUSCHNER, GLANCY, LEVY. 
 
 TABLE XVe Continued. 
 
 Unlt-l" 
 
 
 Sin 
 
 V 
 
 V 
 
 w> 
 
 1* 
 
 + 69+64 
 2-#+69+64 
 #+69+64 
 
 + 7969 
 - 2624 
 - 8819 
 
 - 41736 
 + 12577 
 + 47347 
 
 -111337 
 
 '/' 
 
 - + 25+24 
 -#+20+24 
 -2t+#+20+24 
 
 + 2246 
 - 396 
 - 3596 
 
 - 6168 
 - 1494 
 + 12561 
 
 + 9351 
 
 1* 
 
 2t 
 3t-# 
 
 +# 
 
 + 423 
 + 357 
 - 780 
 
 - 1797 
 - 2207 
 + 4005 
 
 
 f 
 
 2e+ 49+44 
 3t-#+40+44 
 +#+49+4J 
 
 - 1783 
 + 924 
 + 1220 
 
 + 3946 
 - 3327 
 ^ 1026 
 
 
 1* 
 
 2t+ 80+84 
 3 t -#+89+84 
 +#+80+8J 
 
 + 6749 
 - 2247 
 - 7252 
 
 - 44127 
 + 14052 
 + 48051 
 
 
 If' 
 
 4 
 *-#+ 4 
 - +</>+ A 
 
 - 285 
 - 1004 
 + 1574 
 
 + 1210 
 + 5771 
 - 8192 
 
 - 2475 
 
 If* 
 
 49+34 
 t-#+49+34' 
 - *+#+40+3J 
 
 -17218 
 + 4253 
 +20345 
 
 + 56961 
 - 8340 
 - 73031 
 
 - 79400 
 
 fir 
 
 + 29+ J 
 2e-#+2fl+ A 
 #+20+ 4 
 
 - 1429 
 - 523 
 + 2280 
 
 + 6138 
 + 3792 
 - 11302 
 
 + 28347 
 
 if 
 
 t+ 2fl+34 
 2t-j+2d+3J 
 #+20+34 
 
 + 1725 
 - 1003 
 - 1492 
 
 - 3054 
 + 3753 
 + 677 
 
 + 13097 
 
 v 
 
 + 65+5J 
 2t-#+6+5J 
 #+60+5J 
 
 -23773 
 + 7038 
 +25974 
 
 +108605 
 - 28427 
 -122380 
 
 +251019 
 
 >? ^ 
 
 - t+ 29+ 4 
 -#+29+ 4 
 -2+#+20+ J 
 
 - 965 
 - 2068 
 + 3785 
 
 + 3533 
 + 10582 
 - 16928 
 
 + 39011 
 
 v 
 
 2t+ A 
 3-#+ 4 
 +#+ 4 
 
 - 820 
 - 470 
 + 1488 
 
 + 3797 
 + 3185 
 - 7870 
 
 
 ||T 
 
 2t+ 49+34 
 3t- #+49+34 
 +#+49+34 
 
 + 1815 
 - 1181 
 - 853 
 
 - 1190 
 + 3807 
 - 3161 
 
 
 --?' 
 
 2t+ 49+54 
 3t-#+49+54 
 +#+49+54 
 
 + 4294 
 - 1571 
 - 4414 
 
 - 17092 
 + 6629 
 + 17198 
 
 
 ^ 
 
 2+ 89+74 
 3t-#+89+74 
 +#+89+74 
 
 -21544 
 + 6700 
 +22868 
 
 +126397 
 - 37167 
 -136294 
 
 
 I" 
 
 -# 
 
 - .+# 
 
 + 866 
 - 866 
 
 - 4261 
 + 4261 
 
 
 * 
 
 49+24 
 -#+49+24 
 - +#+49+24 
 
 +10682 
 - 1815 
 -12428 
 
 - 28347 
 + 474 
 + 37322 
 
 + 32120 
 
 >!" 
 
 + 29+24 
 2 -#+29+24 
 #+29+24 
 
 - 1498 
 + 1136 
 + 861 
 
 + 450 
 - 4394 
 + 3794 
 
 - 22127 
 
 
 
 m' 
 
MEMOIRS NATIONAL ACADEMY OF SCIENCES. 
 
 XVc Continued. 
 
 [Vol. XIV. 
 
 Unit-l" 
 
 
 Sin 
 
 w 
 
 w 
 
 
 
 1* 
 
 + 60+44 
 
 +17790 
 
 - 69344 
 
 
 ttttll- 
 
 2-0+60+44 
 0+60+44 
 
 - 4675 
 -19046 
 
 + 15200 
 + 77260 
 
 -135954 
 
 9" 
 
 -0+20 
 
 + 1634 
 
 - 7081 
 
 + 16199 
 
 f Ti-il I 
 
 -2+0+20 
 
 - 1634 
 
 + 7081 
 
 
 ," 
 
 2+ 24 
 
 + 328 
 
 ^ 1710 
 
 
 
 3-0+ 24 
 
 + 154 
 
 - 1141 
 
 
 
 
 +0+ 24 
 
 - 591 
 
 + 3420 
 
 1 
 
 ,/ 
 
 2+ 40+44 
 
 - 5879 
 
 + 19019 
 
 
 
 3-0+40+44 
 
 + 2032 
 
 - 7361 
 
 
 
 t+0+40+44 
 
 + 5807 
 
 - 17998 
 
 
 ,/J 
 
 2+ 80+64 
 
 +17340 
 
 - 90064 
 
 
 
 3-0+80+64 
 
 - 5018 
 
 + 24266 
 
 
 
 +0+80+64 
 
 -18102 
 
 + 95820 
 
 
 jJ 
 
 -0 
 
 i- 866 
 
 + 4260 
 
 
 <',"!.' - 
 
 - * + 
 
 + 866 
 
 - 4260 
 
 
 J 3 
 
 40+34-2 
 
 + 609 
 
 - 2958 
 
 + 6763 
 
 
 -0+40+34-2" 
 
 + 232 
 
 - 1656 
 
 
 tWfV - 
 
 - +0+40+34-2 
 
 - 1044 
 
 + 5600 
 
 s ; 
 
 > a 
 
 + 20+24 
 
 - 1760 
 
 + 71S9 
 
 
 
 2 S -0+20+24 
 
 - 331 
 
 + 3096 
 
 
 
 0+20+24 
 
 + 2677 
 
 - 12681 
 
 + 30930 
 
 ; 2 
 
 + 60+54-2" 
 
 + 578 
 
 - 3543 
 
 
 
 2 -0+60+54-2" 
 
 + 10 
 
 - 299 
 
 
 
 0+60+54-2 
 
 - 780 
 
 + 5023 
 
 - 15302 
 
 ;' 
 
 -0+20+4-2 
 
 -2f +0+20+4 -2" 
 
 + 866 
 - 866 
 
 - 4260 
 + 4260 
 
 + 10988 
 
 f 
 
 2 + 4+2 
 
 + 1152 
 
 - 4231 
 
 
 
 3 7ft iti 
 
 +0+ 4+2 
 
 + 98 
 - 1634 
 
 1440 
 + 7081 
 
 
 a 
 
 2+ 40+44 
 
 - 1795 
 
 + 9459 
 
 
 
 3-0+40+44 
 
 + 164 
 
 - 17 
 
 
 
 +0+40+44 
 
 + 2229 
 
 - 12595 
 
 
 j 
 
 2 + 80+74-2 
 
 + 392 
 
 - 2914 
 
 V ,' 
 
 
 3-0+80+74-2 
 
 - 40 
 
 + 194 
 
 
 
 +0+80+74-2 
 
 - 482 
 
 + 3691 
 
 
 
 *+ 0+ 4 
 
 + 47.1 
 
 - 149. 3 
 
 + 186 
 
 
 f-0+ 0+ 4 
 
 + 27.5 
 
 - 111.4 
 
 + 207 
 
 
 -*+0 + 0+ 4 
 
 - 90.4 
 
 + 310. 5 
 
 - 455 
 
 
 ?+ 30+34 
 
 + 216. 1 
 
 - 655. 2 
 
 + 749 
 
 
 |s-0+30+34 
 
 - 58.9 
 
 + 150.7 
 
 - 93 
 
 
 $+0+30+34 
 
 - 229. 3 
 
 + 722. 8 
 
 - 905 
 
 
 | + 50+54 
 
 + 113. 8 
 
 - 499. 2 
 
 + 892 
 
 
 ^-0+50+54 
 
 - 33.5 
 
 + 137. 7 
 
 - 213 
 
 
 fs+0+50+54 
 
 - 118. 2 
 
 + 527. 9 
 
 - 977 
 
 
 
 ^+ 70+74 
 
 + 54.1 
 
 - 310. 2 
 
 + 757 
 
 
 $-0+70+74 
 
 - 16.5 
 
 + 91.3 
 
 - 209 
 
 
 +0+70+74 
 
 - 55.7 
 
 + 322. 3 
 
 - 801 
 
 
 $+ 90+94 
 
 + 24.5 
 
 - 173. 5 
 
 + 537 
 
 
 -^-0+90+94 
 
 7.6 
 
 + 52.7 
 
 - 157 
 
 
 $+0+90+94 
 
 - 25.1 
 
 + 178. 5 
 
 - 559 
 
 
 
 
 m' 
 
 
No. 3.] 
 
 MINOR PLANETS LEUSCHNER, CLANCY. LEVY. 
 
 XVc Continued . 
 
 85 
 
 Unit-l* 
 
 
 Sin 
 
 .. 
 
 . 
 
 <c* 
 
 ' 
 
 _|:^I^+3J 
 
 -1497 
 + 419 
 +1729 
 
 + 4732 
 - 966 
 - 5826 
 
 - 5963 
 + 131 
 + 8424 
 
 1 
 
 -i 4- 0+ 4 
 JE-<H- 0+ A 
 
 -|E+^+ 0+ A 
 
 - 114 
 
 - 224 
 + 436 
 
 + 385 
 + 1006 
 - 1726 
 
 - 548 
 - 2186 
 + 3220 
 
 
 
 \s+ 0+ A 
 
 JE + VH- 0+ A 
 
 - 208 
 - 55 
 + 314 
 
 + 781 
 + 349 
 - 1316 
 
 - 1315 
 - 1026 
 
 + 2641 
 
 TJ 
 
 ?E+ 50+54 
 
 -1366 
 + 420 
 +1480 
 
 + 6114 
 - 1711 
 - 6793 
 
 -113C3 
 + 2548 
 +13254 
 
 1 
 
 $E+ 30+34 
 
 + 108 
 - 85 
 - 19 
 
 - 20 
 
 + 256 
 - 329 
 
 - 847 
 - 395 
 -T 15S6 
 
 1 
 
 $E+ 70+74 
 
 - 922 
 + 292 
 + 975 
 
 + 5348 
 - 1618 
 - 5728 
 
 -13320 
 + 3632 
 +14602 
 
 ' 
 
 ^ ^-{-59-|-5J 
 
 %E~^-y -\-50-\-bJ 
 
 + 172 
 - 74 
 - 133 
 
 - 594 
 + 298 
 + 402 
 
 + 491 
 - 470 
 
 - 42 
 
 ' 
 
 JE +90+94 
 |E ^+90+94 
 f+v''+90+94 
 
 - 541 
 + 174 
 + 564 
 
 + 3856 
 - 1205 
 - 4055 
 
 -12092 
 + 3608 
 +12887 
 
 ' 
 
 if +30+24 
 
 +2041 
 - 431 
 -2290 
 
 - 5080 
 + 499 
 + 6274 
 
 + 4928 
 - 1026 
 - 7597 
 
 rf 
 
 ii^"^? l~ w 
 ~~ w~T"ty~T~ U 
 
 + 384 
 - 384 
 
 - 1410 
 + 1410 
 
 + 2605 
 - 2605 
 
 * 
 
 5* V*i Q\~iA 
 
 - 131 
 + 106 
 + 69 
 
 + 12 
 - 366 
 + 350 
 
 + 717 
 + 772 
 - 1728 
 
 * 
 
 IE ^+50+44 
 
 +2169 
 - 596 
 -2295 
 
 - 8241 
 + 1980 
 + 9008 
 
 +12680 
 - 2086 
 
 -14823 
 
 rf 
 
 E ^+30+44 
 ff +^+30+44 
 
 - 389 
 + 135 
 + 383 
 
 + 1251 
 - 479 
 - 1189 
 
 - 1212 
 + 687 
 + 930 
 
 * 
 
 ft +70+64 
 
 +1550 
 - 457 
 -1609 
 
 - 7940 
 + 2211 
 + 8376 
 
 +17170 
 - 4245 
 -18650 
 
 * 
 
 IE ( +50+64 
 
 - 349 
 + 113 
 + 352 
 
 + 1665 
 - 543 
 - 1678 
 
 - 3127 
 + 1052 
 + 3117 
 
 rf 
 
 \t +90+84 
 \i (^+90+84 
 6 -)-^+90-(-84 
 
 + 937 
 - 286 
 - 963 
 
 - 6044 
 + 1784 
 + 6274 
 
 +16950 
 - 4724 
 -17880 
 
 "' 
 
 i +0+4 
 
 ?-^+ 0+ 4 
 
 + 757 
 + 514 
 -1583 
 
 - 3272 
 - 3644 
 + 8214 
 
 
 r i 
 
 is +50+54 
 
 * i.'~|~oi7~i~oj 
 
 "^ ^~ }~ v i" OW~4~O J 
 
 +7767 
 -2522 
 -8820 
 
 -35692 
 +10085 
 +42033 
 
 
 
 
 
 
MEMOIRS NATIONAL ACADEMY OF SCIENCES. 
 
 TABLE XVc Continued. 
 
 [Vol. XIV. 
 
 Unlt-l" 
 
 
 Sin 
 
 Uio 
 
 w 
 
 
 
 f 
 
 -i +35+34 
 is-0+30+34 
 -}+0+30+34 
 
 + 4758 
 - 1369 
 - 6128 
 
 - 15945 
 + 2307 
 + 22836 
 
 
 1* 
 
 $ +39+34 
 $-0+30+34 
 is+0+30+34 
 
 - 882 
 + 732 
 + 177 
 
 - 280 
 - 2580 
 + 3816 
 
 
 
 1J' 
 
 ft +70+74 
 jf-0+70+74 
 is+0+70+74 
 
 + 7549 
 - 2504 
 - 8212 
 
 - 44427 
 + 13864 
 + 49192 
 
 
 ? 
 
 -V + 8+ A 
 
 -i-0+ 0+ ^ 
 
 -|+0+ + 4 
 
 - 32 
 
 + 784 
 - 1031 
 
 + 220 
 - 4194 
 + 5051 
 
 
 f 
 
 $ +99+94 
 \s- 0+90+94 
 $+0+90+94 
 
 + 5780 
 - 1929 
 - 6154 
 
 - 41583 
 + 13412 
 + 44735 
 
 
 v 
 
 i +0 
 
 $-0+ 9 
 -i+^+ 
 
 - 768 
 - 1156 
 + 1924 
 
 + 2821 
 + 6406 
 - 9227 
 
 
 ty 
 
 4 + 0+2J 
 |e-0+ 9+2J 
 . -l+^+ 0+2J 
 
 + 209 
 - 771 
 + 446 
 
 + 1404 
 + 3774 
 - 5879 
 
 
 ,,' 
 
 J +50+4J 
 i -^+50+4J 
 -i+^+50+4J 
 
 -21869 
 + 6125 
 +24564 
 
 + 85960 
 - 19358 
 -101260 
 
 
 ,,' 
 
 -J +30+2J 
 i-^+30+2J 
 -i +^+30+2J 
 
 -10182 
 + 1576 
 +13528 
 
 + 27638 
 + 2276 
 - 43309 
 
 
 y 
 
 \t +30+24 
 4-t^+30+2J 
 j+^+30+2J 
 
 + 384 
 - 939 
 + 715 
 
 + 3626 
 + 3382 
 - 8550 
 
 
 V 
 
 f +30 +4 A 
 *-^+30+4J 
 i+0+30+44 
 
 + 3250 
 - 1333 
 - 3256 
 
 - 10017 
 + 4996 
 + 9153 
 
 
 iV 
 
 f +70+6J 
 it- ^+70+64 
 j+^+70+6J 
 
 -23414 
 + 7146 
 +25145 
 
 +122108 
 - 34410 
 -133985 
 
 
 * 
 
 -ft + 
 
 -J -V>+ 
 
 -| + ^+ 
 
 + 768 
 - 2308 
 + 1540 
 
 - 2821 
 + 10637 
 - 7816 
 
 
 ili 
 
 | +90+84 
 ^-^+90+8J 
 $+0+90+84 
 
 -18847 
 + 5935 
 +19837 
 
 +122928 
 - 37138 
 -130949 
 
 
 i" 
 
 i +0+4 
 
 |-0+ 0+ 4 
 
 -l+^+ 0+ 4 
 
 + 761 
 + 906 
 - 1920 
 
 - 3333 
 
 - 5387 
 + 9831 
 
 
 ," 
 
 is +50+34 
 
 $-^+50+34 
 -i+0+50+34 
 
 +15303 
 - 3577 
 -16828 
 
 - 49954 
 + 7957 
 + 58649 
 
 
 ^ 3 
 
 -Jf +30+ 4 
 i-0+30+ 4 
 -f+0+30+ 4 
 
 + 1582 
 + 1300 
 - 3410 
 
 - 5765 
 - 6572 
 + 14260 
 
 
 v a 
 
 \i -0+4 
 
 |-0- 0+ 4 
 
 is+0- 0+ 4 
 
 + 451 
 + 494 
 - 1096 
 
 - 1890 
 - 2861 
 + 5381 
 
 
 
 
 mf 
 
No. 3.] 
 
 MINOR PLANETS LEUSCHNER, GLANCY, LEVY. 
 
 87 
 
 TABLE XVc Continued. 
 T, 
 
 Unit-l" 
 
 
 Sin 
 
 - 
 
 w 
 
 *. 
 
 1* 
 
 f +30+34 
 
 - 3918 
 
 + 8760 
 
 
 
 is ^+30+34 
 
 + 1588 
 
 - 5289 
 
 
 
 $+VH-30+34 
 
 + 3637 
 
 - 6391 
 
 
 l" 
 
 ft +70+54 
 
 + 18292 
 
 - 83098 
 
 
 
 |j-^+70+54 
 
 - 5104 
 
 + 20825 
 
 
 
 it+^+70+54 
 
 - 19286 
 
 + 89973 
 
 
 f 
 
 * +0+4 
 
 - 902 
 
 + 3781 
 
 
 
 $!-</>+ 0+ 4 
 
 - 988 
 
 + 5721 
 
 
 
 -i+#+ 0+ 4 
 
 + 2191 
 
 - 10762 
 
 
 jl 
 
 J +50+44 - 1 
 
 + 634 
 
 - 3482 
 
 
 
 fa ^-j-50+44 jf 
 
 + 87 
 
 - 836 
 
 
 
 -j+^+50+44-.2f 
 
 - 933 
 
 + 5479 
 
 
 f 
 
 -it +30+24-J 
 
 + 428 
 + 480 
 
 - 1816 
 - 2805 
 
 
 
 -5H-VH-30+24-1 
 
 - 1050 
 
 + 5226 
 
 
 p 
 
 ft +30+34 
 
 - 1916 
 
 + 8929 
 
 
 
 it ^+30+34 
 
 + 2 
 
 + 1220 
 
 
 
 it+^+30+34 
 
 + 2553 
 
 - 13126 
 
 
 f 
 
 Jf +70+64 -S 
 
 + 488 
 
 - 3307 
 
 
 
 I _^-|- 70+64 2" 
 
 27 
 
 + 23 
 
 
 
 $+^+70+64 -2 
 
 - 623 
 
 + 4387 
 
 
 p 
 
 -* +0 -J 
 
 - 475 
 
 + 1965 
 
 
 
 55 ^+ JT 
 
 + 1141 
 
 - 5536 
 
 
 
 -i+0+ -2" 
 
 - 508 
 
 + 2916 
 
 
 p 
 
 it + 0+24+J 
 
 + 1282 
 
 - 5447 
 
 
 
 $-<+ 0+24+J 
 
 - 90 
 
 384 
 
 
 
 | +^+ 0+24+J 
 
 - 1620 
 
 + 7647 
 
 
 p 
 
 is +50+54 
 
 - 1544 
 
 + 9111 
 
 
 
 ^ ^+50+54 
 
 + 222 
 
 - 735 
 
 
 
 f+^+50+54 
 
 + 1838 
 
 - 11413 
 
 
 p 
 
 | t + 9g +8 j_j 
 
 + 304 
 
 - 2460 
 
 
 
 Jt ^+90+84 J 
 
 - 42 
 
 + 266 
 
 
 
 fj+^+90+84 J 
 
 - 364 
 
 + 3013 
 
 
 ,1 
 
 20+24 
 
 - 1955 
 
 + 14862 
 
 
 
 60+64 
 
 - 35276 
 
 + 189348 
 
 
 
 9 
 
 + 3312 
 
 - 23724 
 
 
 
 df~\~A.Q ~j~4d 
 
 - 5097 
 
 - 4328 
 
 
 
 ^+40+44 
 
 + 6177 
 
 - 16310 
 
 
 
 C& ~f o V ~|~ 8 a 
 
 + 45199 
 
 - 304998 
 
 
 iV 
 
 26+ 4 
 
 + 6733 
 
 - 33547 
 
 
 
 20+34 
 
 - .3730 
 
 + 1693 
 
 
 
 60+54 
 
 +142854 
 
 - 673242 
 
 
 
 <1> +4 
 
 - 9270 
 
 + 61512 
 
 
 
 -# +4 
 
 + 4207 
 
 - 28940 
 
 
 
 V^+40+34 
 
 + 5323 
 
 + 55061 
 
 
 
 -i+40+34 
 
 - 13730 
 
 + 9080 
 
 
 
 V^+40+54 
 
 + 22898 
 
 - 84425 
 
 
 
 V^+80+74 
 
 -200024 
 
 +1218446 
 
 
 7 ?" 
 
 20 
 
 - 3268 
 
 + 14164 
 
 
 
 20+24 
 
 + 3445 
 
 + 15177 
 
 
 
 60+44 
 
 -190467 
 
 + 772593 
 
 
 
 
 
 + 12782 
 
 - 78712 
 
 
 
 ^ +24 
 
 + 5239 
 
 - 35125 
 
 
 
 ^+40+24 
 
 + 2712 
 
 - 60586 
 
 
 
 -Ji+40+24 
 
 + 4409 
 
 + 41693 
 
 
 
 ^+40+44 
 
 - 52183 
 
 + 143461 
 
 
 
 V-+80+64 
 
 +294332 
 
 -1600036 
 
 
 
 
 m' 
 
88 
 
 MEMOIRS NATIONAL ACADEMY OF SCIENCES. 
 
 [Vol. XIV. 
 
 TABLE XVc Continued. 
 
 Unit-l" 
 
 
 Sin 
 
 >o 
 
 w 
 
 
 
 r/ 3 
 
 26+ A 
 
 + 3479 
 
 - 17883 
 
 T 
 
 
 65+34 
 
 + 83314 
 
 - 283500 
 
 
 
 <!> + 4 
 
 - 7839 
 
 4- 47423 
 
 
 
 -<l>+46+ A 
 
 + 6634 
 
 - 36904 
 
 
 
 V>+40+34 
 
 + 27512 
 
 - 44330 
 
 
 
 0+80+54 
 
 -144023 
 
 + 688658 
 
 
 fr, 
 
 25+24 
 
 + 10709 
 
 - 50725 
 
 
 
 2(9+ A -2 
 
 - 1732 
 
 + 8521 
 
 
 
 65+54 -.J 1 
 
 - 7799 
 
 + 50227 
 
 
 
 ^ 
 
 - 12782 
 
 + 78712 
 
 
 
 # + 4+JT 
 
 + 11006 
 
 - 60629 
 
 
 
 4>+45+34-.T 
 
 + 4022 
 
 - 29208 
 
 tV 
 
 
 -0+45+34 -2 1 
 
 - 3616 
 
 + 27235 
 
 
 
 0+45+44 
 
 - 28408 
 
 + 176052 
 
 
 
 0+85+74-2 1 
 
 + 9526 
 
 - 75678 
 
 
 f if 
 
 28+ A 
 
 - 7475 
 
 + 36068 
 
 
 
 26+24-2 
 
 + 159 
 
 - 2967 
 
 
 
 65+44 -J 
 
 + 11564 
 
 - 66719 
 
 
 
 +4 
 
 + 11762 
 
 - 75153 
 
 
 
 ^ +^ 
 
 - 6024 
 
 + 38182 
 
 
 
 -0+45+24 -.T 
 
 + 7090 
 
 - 45771 
 
 
 
 0+45+34 
 
 + 35006 
 
 - 199168 
 
 
 
 0+45+44 -I 
 
 + 1108 
 
 - 281 
 
 i. 
 
 
 0+80+64-.? 
 
 - 15308 
 
 + 111481 
 
 
 
 
 m' 
 
 An inspection of the preceding table, which is typical of all the trigonometric series under 
 consideration, shows readily that any function of this type is of the form 
 
 lie' sin K' + lJcsin (K</>) = 2k' sin K' + Ilcsin K cos d> +_ lie cos K sin </ 
 or 
 
 IV cos K' + Ik cos (K<{>)=21c' cos K' + Ilc cos Jf cos ^Ilc sin if sin <j> 
 
 or, more briefly, a + 6 cos ^ + c sin ^ 
 
 wherea, 6, care trigonometric series and can be written by inspection from the tabulated function. 
 Hence, in v. Zeipel's notation (Z 54, eq. 96), 
 
 T { = X { + Y { cos <l> + Z t sin <J> 
 and the integral may be written 
 
 W^ = x^+y^ cos ^+2< sin <[> 
 
 The functions T and W are to be used in this form in solving equations (95). 
 Considering only first order in the mass in T 
 
 T 2 = X 2 + Y 2 cos ([> + Z 2 sin ^ 
 where 
 
 X 2 = Ilc' sin K'; Y 2 = Ik sin K; Z 2 = Ik cos K 
 
 or, X 2 is the part of T 2 which is independent of </>, Y 2 is a trigonometric sine series having the 
 same numerical coefficients as the part of T 2 which contains <[> in the argument, but in which 
 <f> is omitted from the argument, and Z 2 is the corresponding cosine series. 
 
 Considering the first two of the eqs. (95), the first one states that W 1 is not a function of 
 alone, or, W FW1 = n- W=rTF1 
 
 "l L iJ ", "1 L "iJ- t2+W 
 
 Making use of this fact in the second, Wj can be obtained from (95 2 ). (See Z 54.) Introducing 
 the auxiliary functions^ and u v defined by (99) and (101), the differential equation for TFj is 
 replaced by the equivalent differential equations, (100) and (102), for ip^ and u v . 
 The series 
 
 and 
 
 can be written by inspection from T 2 , or, better, the integration itself can be performed in part 
 at the same time. 
 
NO. 3.] MINOR PLANETS LEUSCHNER, GLANCY, LEVY. 89 
 
 The function fa is given by Z 59, eq. (103), or, 
 
 From the table of T 2 , page 82, it is not difficult to write immediately 
 
 
 The terms of higher order must be obtained by the usual method for the mechanical multi- 
 plication of series. A logarithmic multiplication is the most direct. 
 
 In each term in the expression for fa the terms of lowest rank must be of the first rank. 
 
 TfL *??? f ffL 
 
 Recalling the tabulation of factors in Z 53, w, , -^> -^> etc., are all of first rank. But the 
 
 coefficient for a given argument consists of three terms in ascending powers of w. Hence 
 fa w, within the limits of the given tabulation for T 2 , is of rank 1, 2, 3 for each order in the 
 mass. Table XVI, giving fa w, is tabulated with double headings. The three subheadings 
 indicate the expansion of the coefficients in a Taylor's series and the main headings give the 
 factors in the development of the radical in Z 59, eq. (103). 
 
 Having found fa, its reciprocal, fa- 1 , inclusive of first order in the mass, is given by 
 
 The second term is the negative of the first three columns of Table XVI multiplied by tff-*. 
 
 QAL 
 
 The product of 2 fa- 1 and that part of T t which contains <p gives -^, and integration with respect 
 
 to 6 gives u v tabulated in XVIII. The function u t is of first and higher rank because the factor 
 fa- 1 is of rank minus one and T 2 is of second rank. 
 
 From Table XVHI y l can be read by inspection, and ijy 1 added to Table XVI gives x lr 
 tabulated in Table XVII. The function W l is the sum of Tables XVII and XVIII. 
 
 In the integration those terms whose arguments are independent of are of the nature of 
 constants. In accordance with the condition that there may be secular terms in 0, the integral 
 contains such terms as the following: 
 
 '* _ _ 0-fcsin 
 
 As the constant of integration 
 
 -ifc sin 
 
 - 
 is added. Hence the integral contains terms such as 
 
 w 
 
 where i s the value of 6 for the time t = 0. 
 
 In passing, it should be noted that, in order that the expansion of Z 59, eq. (103), shall 
 represent the function, we must have 
 
 - t 4* V? 
 
 - i 
 
 - . u . W> 
 
 and this condition should be tested for a given planet before applying this method of determining 
 the perturbations. 
 
 To the computer the extent of auxiliary tables, the arrangement of series in logarithms or 
 natural numbers, in seconds of arc or radians, inclusive or exclusive of numerical factors, and 
 foresight in combining operations all these are of the greatest importance. But considerations 
 of this kind would carry the reader into complicated details which are best left to the com- 
 puter's own judgment. 
 
 On the other hand, general considerations about the extent of the published tables are of 
 importance in the discussion of the accuracy of the final tables. Yet, for a given limit of 
 accuracy, it is so difficult to determine, for each table, the highest powers of m', w, TJ, T)', and j 2 
 that little or nothing is said about it in connection with individual tables, but the discussion 
 is reserved until later. 
 
90 
 
 MEMOIRS NATIONAL ACADEMY OF SCIENCES. 
 
 [Vol. XIV. 
 
 TABLE XVI. 
 <t >l -w=x l -i 1 y l =[(l-e COB 
 
 Unit 4th decimal of a radian. 
 
 
 Prw 
 
 10-1 
 
 )- 
 
 to-* 
 
 
 vm 
 
 w 
 
 w 
 
 w* 
 
 w" 
 
 w 
 
 w' 
 
 w' 
 
 w 
 
 V* 
 
 
 
 
 
 -0. 0460 
 
 +0. 231 
 
 -0.52 
 
 
 
 ^ 
 
 
 
 
 
 -0. 0060 
 
 +0. 040 
 
 -0. 127 
 
 
 
 *r 
 
 4 
 
 
 
 
 +0. 0331 
 
 -0. 195 
 
 +0.53 
 
 
 
 ij 
 
 25+24 
 
 + 42. 889 
 
 - 107. 72 
 
 + 106.7 
 
 
 
 
 
 
 ' 
 
 25+ 4 
 
 - 15. 427 
 
 + 52. 39 
 
 - 75.2 
 
 
 
 
 
 
 i f 
 
 45+44 
 
 - 122.10 
 
 + 484.1 
 
 - 813 
 
 -0. 0460 
 
 +0. 231 
 
 -0.52 
 
 
 
 9< 
 
 45+34 
 
 + 357. 75 
 
 - 1183.5 
 
 +1650 
 
 +0. 0331 
 
 -0. 195 
 
 +0.53 
 
 
 
 *" 
 
 45+24 
 
 - 258.93 
 
 + 687. 2 
 
 - 779 
 
 -0. 0060 
 
 +0. 040 
 
 -0. 127 
 
 
 
 ; 
 
 45+34 -.T 
 
 - 14. 75 
 
 + 71.7 
 
 - 164 
 
 
 
 
 
 
 l 1 
 
 25+24 
 
 + 28.2 
 
 - 433 
 
 
 +0. 262 
 
 -1.70 
 
 
 +0.0003 
 
 -0. 0022 
 
 V 
 
 65+64 
 
 + 428 
 
 - 2295 
 
 
 +0. 262 
 
 -1.70 
 
 
 +0. 0001 
 
 -0.0008 
 
 5V 
 
 25+ 4 
 
 - 316.1 
 
 + 1592 
 
 
 -0. 767 
 
 +4.46 
 
 
 -0. 00021 
 
 +0. 0018 
 
 ,y 
 
 25+34 
 
 + 108.5 
 
 - 49 
 
 
 -0. 094 
 
 +0.69 
 
 
 -0. 00011 
 
 +0.0009 
 
 ,y 
 
 65+54 
 
 -1889 
 
 + 8902 
 
 1 t , 
 
 -0.86 
 
 +5.2 
 
 
 -0. 0001 
 
 +0.001 
 
 -n" 
 
 25 
 
 + 237.6 
 
 - 1030 
 
 
 +0. 555 
 
 -2.87 
 
 
 +0. 00004 
 
 -0.0002 
 
 ,," 
 
 25+24 
 
 - 125.3 
 
 - 552 
 
 
 +0. 276 
 
 -1.85 
 
 
 +0. 00008 
 
 -0.0009 
 
 ,," 
 
 65+44 
 
 +2770 
 
 -11237 
 
 
 +0.83 
 
 -4.7 
 
 
 
 
 ," 
 
 25+ 4 
 
 - 168.7 
 
 + 867 
 
 
 -0.200 
 
 +1.21 
 
 
 
 
 ,"> 
 
 65+34 
 
 -1346 
 
 + 4581 
 
 
 -0.200 
 
 +1.21 
 
 
 
 
 ft 
 
 25+24 
 
 - 389.4 
 
 + 1846 
 
 -4498 
 
 
 
 
 
 
 ft 
 
 25+ 4-2 1 
 
 + 126.0 
 
 - 620 
 
 
 +0. 032 
 
 -0.23 
 
 
 
 
 ft 
 
 65+54 -J 
 
 + 113 
 
 - 731 
 
 
 +0. 032 
 
 -0.23 
 
 
 
 
 ? rf 
 
 25+ 4 
 
 + 362. 4 
 
 - 1749 
 
 
 
 
 
 
 
 p if 
 
 25+24 -JT 
 
 - 7.7 
 
 + 144 
 
 
 -0. Oil 
 
 +0.09 
 
 
 
 
 }> (' 
 
 65+44 -.T 
 
 -i 187 
 
 + 1078 
 
 
 -0. Oil 
 
 +0.1 
 
 
 
 
 
 
 m' 
 
 mf 
 
 m" 
 
 TABLE XVII. 
 
 Unit-l' 
 
 
 
 UJ I II}- 1 
 
 
 PrtO 
 
 
 
 I/OS 
 
 w* 
 
 w 
 
 UI 
 
 w' 
 
 w 
 
 U) 
 
 1) 
 
 25+24 
 
 + 1179.6 
 
 - 2963 
 
 + 2935 
 
 
 
 
 V 
 
 25+ 4 
 
 - 318. 2 
 
 + 1081 
 
 - 1552 
 
 
 
 
 * 
 
 
 
 
 
 - 0.95 
 
 + 4.8 
 
 
 I 3 
 
 45+44 
 
 - 3358 
 
 + 13313 
 
 - 22356 
 
 - 1.27 
 
 + 6.4 
 
 
 n' 
 
 4 
 
 
 
 
 + 0.68 
 
 - 4.0 
 
 
 W 
 
 45+34 
 
 + 8609 
 
 - 28481 
 
 + 39702 
 
 + 0.79 
 
 -4.7 
 
 
 ^ 
 
 
 
 
 
 - 0.12 
 
 + 0.8 
 
 
 v 
 
 45+24 
 
 - 5341 
 
 + 14175 
 
 - 16063 
 
 - 0.12 
 
 + 0.8 
 
 
 ? 
 
 45+34 -2 
 
 - 304 
 
 + 1479 
 
 - 3383 
 
 
 
 
 t 
 
 25+24 
 
 + 1955 
 
 - 14861 
 
 
 + 7.2 
 
 - 46.6 
 
 
 ?' 
 
 65+64 
 
 +11758 
 
 - 63112 
 
 
 + 7.2 
 
 - 46.6 
 
 
 ,y 
 
 25+ 4 
 
 - 6732 
 
 + 33547 
 
 
 -15.2 
 
 + 88.0 
 
 
 ,y 
 
 25+34 
 
 + 3730 
 
 - 1691 
 
 
 -3.8 
 
 + 27.9 
 
 
 ,y 
 
 65+54 
 
 -47616 
 
 +224423 
 
 
 -21.7 
 
 +130. 
 
 
 ,," 
 
 25 
 
 + 3267 
 
 - 14165 
 
 
 + 7.4 
 
 - 37.8 
 
 
 ,," 
 
 25+24 
 
 - 3446 
 
 - 15176 
 
 
 + 7.8 
 
 - 52.5 
 
 
 ??* 
 
 65+44 
 
 +63489 
 
 -257533 
 
 
 +19.0 
 
 -108. 1 
 
 
 ft 
 
 25+ 4-2 1 
 
 + 1733 
 
 - 8522 
 
 
 + 0.4 
 
 - 3.1 
 
 
 ft 
 
 65+5 J-2 
 
 + 2599 
 
 - 16744 
 
 
 + 0.7 
 
 - 5.2 
 
 
 ft 
 
 25+24 
 
 -10709 
 
 + 50748 
 
 -123705 
 
 
 
 
 ," 
 
 25+ 4 
 
 - 3479 
 
 + 17880 
 
 
 - 4.1 
 
 + 24.9 
 
 
 I? 
 
 65+34 
 
 -27772 
 
 + 94500 
 
 
 - 4.1 
 
 + 24.9 
 
 
 ? i 
 
 25+24 -JT 
 
 - 159 
 
 + 2966 
 
 
 - 0.2 
 
 + 1.9 
 
 
 f ri' 
 
 65+44-2' 
 
 - 3855 
 
 + 22240 
 
 
 - 0.2 
 
 + 1.9 
 
 
 ? 1 
 
 25+ J 
 
 + 7475 
 
 - 36070 
 
 
 
 
 
 
 (0-5 )sin 
 
 
 
 
 
 
 
 W' 
 
 A 
 
 - 570 
 
 + 2421 
 
 4950 
 
 - 0.45 
 
 + 2.7 
 
 -7.2 
 
 
 
 m' 
 
 m" 
 
No. 3.] 
 
 MINOR PLANETS LEUSCHNER, CLANCY, LEVY. 
 
 . ..,,. TABLE XVIII. 
 
 91 
 
 j=y, coe 
 
 An </> 
 
 Unit-l". 
 
 
 COB 
 
 M 
 
 m-> 
 
 tr V 
 
 ^ 
 
 * 
 
 " 
 
 V 
 
 V 
 
 <J>+2d+2J 
 
 + 294.89 
 - 839.5 
 + 1229.8 
 
 - 740.6 
 + 3328 
 - 4069 
 
 + 734 
 - 5586 
 + 5671 
 
 - 0. 316 
 + 0. 114 
 
 + L59 
 - 0.67 
 
 - 3.6 
 + L8 
 
 r 
 
 -J+20+2J 
 
 + 396 
 + 978 
 + 2940 
 
 + 1494 
 - 7431 
 - 15782 
 
 - 9351 
 + 23105 
 [+ 37112] 
 
 - 2.62 
 + 4.42 
 + 1.80 
 
 + 16.8 
 - 28.4 
 - 1L7 
 
 
 li* 
 
 -++29+ J 
 
 V>+20+ A 
 
 + 2068 
 + 1492 
 - 2280 
 - 8658 
 
 - 10582 
 - 677 
 + 11302 
 + 40793 
 
 - 39010 
 - 13058 
 - 28348 
 - 83730 
 
 + 6.18 
 - L91 
 - 5.57 
 -3.95 
 
 - 36.9 
 + 13.6 
 
 + 32.8 
 + 23.6 
 
 
 e 
 
 <j>+26+2J 
 +60+4.1 
 
 - 1634 
 - 861 
 + 6349 
 
 + 7081 
 - 3794 
 - 25753 
 
 - 16199 
 + 22127 
 + 45318 
 
 - 4.04 
 + 2.12 
 + L90 
 
 + 2L4 
 - 14.4 
 - 10.8 
 
 
 i 
 
 -4+20+ J-J 
 
 J+26+2J 
 
 - 866 
 + 260 
 - 2677 
 
 + 4260 
 - 1674 
 + 12681 
 
 - 10988 
 + 5101 
 - 30930 
 
 - 0.22 
 + 0.07 
 
 + L6 
 - 0.5 
 
 
 I 
 
 0+4J+4J 
 <p+46+4J 
 $5+80+8.1 
 
 + 2549 
 - 3089 
 -LI 300 
 
 + 2164 
 + 8155 
 + 76250 
 
 
 F-1L9J 
 
 'ill 1 
 
 + 89 
 - 25 
 + 70 
 
 
 i 
 
 rr 
 
 _^-f-40-)_3J 
 ^+80+74 
 
 -11449 
 - 2661 
 + 6865 
 +50005 
 
 + 42212 
 - 27530 
 - 4540 
 -304611 
 
 
 + L9 
 [+36.4] 
 -20.3 
 [+33.8] 
 
 - 23 
 
 -241] 
 +118] 
 -248] 
 
 
 if 
 
 ^-|-40-|-4j 
 VH-40+2J 
 
 +26091 
 - 1356 
 - 2204 
 -73583 
 
 - 71730 
 + 30293 
 - 20846 
 +400009 
 
 
 -10.1 
 
 f-25.51 
 1+28.01 
 1-4L9] 
 
 + 83 
 +153 
 -153 
 
 +284 
 
 
 ^ 
 
 -fjfj^ 
 
 -13756 
 - 3317 
 
 +36006 
 
 + 22165 
 + 18452 
 -172164 
 
 
 +10.1 
 -12.4 
 +16.6 
 
 - 65 
 + 64 
 -104 
 
 
 i 
 
 111^ 
 
 - 2011 
 + 1808 
 - 2381 
 +14204 
 
 + 14604 
 - 13617 
 + 18919 
 - 88026 
 
 
 - L9 
 
 'it! 1 
 
 + 5.7 
 
 + 14 
 [- 14] 
 
 + 10 
 
 [-42] 
 
 
 H 
 
 -#+40+2J-J 
 
 - 554 
 - 3545 
 
 + 3827 
 -17503 
 
 + 140 
 + 22886 
 - 27870 
 + 99584 
 
 
 + 0.5 
 - L8 
 + L3 
 [- 3.7] 
 
 - 4 
 + 14 
 - 11 
 [+28] 
 
 
 , 
 
 ,5 08U1 
 
 + 767. 72 
 
 - 2820.9 
 
 + 5210 
 
 + L265 
 
 - 6.35 
 
 +14.3 
 
 V 
 
 v> + J 
 
 - 569.95 
 
 + 2421.1 
 
 - 4950 
 
 - 0.455 
 
 + 2.69 
 
 -7.2 
 
 
 
 t 
 
 + 6624 
 
 - 47448 
 
 
 +23.8 
 
 [-22L 9] 
 
 
 ?Y 
 
 t + J 
 
 [-18540] 
 + 8414 
 
 [+123024] 
 
 - 57880 
 
 
 -73.4 
 +36.0 
 
 +572.4 
 -282.2 
 
 
 ' 
 
 J +2J 
 
 +10478 
 +25564 
 
 - 70250 
 -157424 
 
 
 +55.2 
 +87.3 
 
 -374.8 
 -652.8 
 
 
 ** 
 
 !> + J 
 
 -15678 
 
 + 94846 
 
 
 -69.9 
 
 +438.6 
 
 
 
 
 /*! 
 
 t 
 
 +22012 
 -25564 
 
 -121258 
 +157424 
 
 +359162 
 [-511232] 
 
 + 9.9 
 -23.1 
 
 - 77.0 
 +165.0 
 
 
 ? 
 
 v +2 
 
 -12048 
 +23524 
 
 + 76364 
 -150306 
 
 -251640 
 
 +498328 
 
 - 5.2 
 +14.8 
 
 + 45.8 
 -112.0 
 
 
 
 m' 
 
 m' 2 
 
92 MEMOIRS NATIONAL ACADEMY OF SCIENCES. 
 
 After the determination of W,, the function F 2 [FJ is obtained from the solution of 
 Z 53, eq. (95 2 ). The integral may be written as in Z 63, eqs. (105), (106), or, quite as simply,, 
 as follows: 
 
 F 2 ' = - a- cos e )(w+ F t ) -[(l -e cos 
 
 The function F 2 ' is given in Table XIX. 
 
 Anticipating some later developments, for which we shall need 
 
 [(l-e cos c)F] 
 the function 
 
 [(l-ecosOF,'] 
 is tabulated in Table XX. 
 
 The determination of [ F 2 ] may be accomplished according to Z 65, eq. (108) Z 67, eq. (116), 
 or in the manner outlined below, which we regard as preferable. 
 Repeating Z 65, eq. (107), 
 
 in which all the known parts are contained on the right-hand side, the development of equivalent 
 equations proceeds in a manner analogous to that for TT,. 
 Writing 
 
 T 3 = X 3 + Y 3 cos <J> + Z 3 sin t}> 
 and introducing 
 
 and equating parts independent of 0, coefficients cf cos <j> and coefficients of sin </>, the three 
 equivalent equations are: 
 
 [(1 - J cos )(;+ FO^J^-t^J)^]- [(1 - 
 
 e cos 
 
 ~ e cos 
 
 1 Vir- 1 - M^i 
 
 i ~3 - A ] 1+ 9 fll )Jw 
 
 [(1 -e cos e ) (w+ F^^J^-tZJ)^]- [(1 -e cos ) J ^[2 
 Multiplying the second of these by ij and subtracting from the first: 
 
 e cos 
 
 [(l- 
 
 [(1-6 cos )J(T 2 -[T 2 ])deJj+2[X 3 -r 1 Y 3 \- 
 
MINOR PLANETS LEUSCHNER, GLANCY, LEVY. 93 
 
 Multiplying the second by cos $, the third by sin tf> and adding: 
 A( tt _ WU)= _+(l-e C03 s)F l -r i F 1 + l + 2([yj cos ^ + [ZJ sin f) 
 
 F,)^ f { F, cos ^ + Z, sin v'--[F, cos 
 
 (1 -e cos )(+ ,) , cos ^ + , sn v'--, cos +, sn 
 
 in which 
 
 = [yj cos Hz sn 
 
 and [.XVI, [FJ, [Z 3 ] are read by inspection from T 3 , which is to be determined as follows: 
 If Z 50, eqs. (89), (90), are written in the form 
 
 _t i 
 
 -= [1 cos(/ w)] = 4U 2r cos e cos (s <!>)+.i) cos (2s ^)-f-ij cos t& + 
 
 cos 1 9> I 
 
 9/v*_ 1 
 
 1 n* 8 n cos s + 2 7/ 1 cos 2 2 cos ( A) 
 
 -8 if cos (e-^)+2ij cos (2- 
 then Tw and T' r , given by Z 49, eqs. (84), (85), in connection with Z 50, eq. (87), are given by 
 
 2V=-r,-4{l-2jj cose-cos (s-^)+ij cos (2e-v'')+7 cos <f>+ ____ } 
 IS P . ,(n + r.-n+s)iji^'/ 2 ' sin A 
 
 r = {3 + 14^-8^ cos s + 2jj cos 2s-? cos (j-^-Sij 1 cos (s-^)+2i9 cos (2s-^)+2ij cos ^ 
 
 o 
 
 and r, (Table X\Tna) is computed by Z 53, eq. (94), in which 
 
 S 
 The function 
 
 is tabulated in Table XXI; the function 
 
 u = [/] cos 
 
 is tabulated in Table XXII. 
 
 From the latter [?/,] can be read by inspection, and ijfyj added to the former gives [xj. 
 Finally, (Table XXIIa), 
 
 F W t ] = [rj + [y J cos ^ + [zj shi ^ 
 
 > * . : M M ^ 
 
 L;. 
 
94 
 
 MEMOIRS NATIONAL ACADEMY OF SCIENCES. 
 TABLE XVIIIa. 
 
 [Vol. XIV. 
 
 Unit=l" 
 
 
 Qin 
 
 
 ur* 
 
 
 
 r-> 
 
 
 
 oin 
 
 w* 
 
 w 
 
 w' 
 
 to 
 
 w 
 
 Vfl 
 
 
 -+# 
 
 
 
 
 +0. 339 
 
 - 2.01 
 
 
 
 < +20+24 
 2t-v&+20+24 
 j+26+24 
 
 
 
 
 -0. 375 
 -0. 137 
 
 +0. 498 
 
 + 2.403 
 + 0. 847 
 - 3.223 
 
 + 7.72 
 
 
 2i +40+44 
 t+^+40+44 
 
 
 
 
 -0. 438 
 +0. 429 
 
 + 2. 234 
 - 2.338 
 
 
 
 2t+4>+60+64 
 
 
 
 
 +0. 361 
 
 - 2.372 
 
 
 
 20+24 
 f-H-20+24 
 - r+^+20+24 
 
 -0. 00047 
 
 +0. 0036 
 
 -0. 0123 
 
 +2. 199 
 +0. 286 
 -3. 294 
 
 -14.58 
 - 3.85 
 +23.82 
 
 + 33. 34 
 
 
 t 
 # 
 
 -0. 00038 
 
 +0. 0035 
 
 -0. 0136 
 
 -2. 811 
 -0. 688 
 
 +12. 20 
 + 1.67 
 
 - 16.51 
 
 ! 
 
 e +40+44 
 0+40+44 
 
 -0. 00015 
 
 +0. 0013 
 
 -0.0048 
 
 +0. 432 
 -4. 536 
 
 - 2.58 
 +35. 80 
 
 - 95.79 
 
 3 
 
 e+0+20+24 
 
 
 
 
 +1.017 
 
 - 6. 333 
 
 
 9 
 
 t+^+60+64 
 
 
 
 
 -3. 219 
 
 +22. 43 
 
 
 \ 
 
 25+ 4 
 I-0+20+ 4 
 - +0+20+ 4 
 
 +0. 00017 
 
 -0. 0014 
 
 +0. 0055 
 
 -2.520 
 -1. 253 
 
 +4. 372 
 
 +14. 78 
 +10. 20 
 -28. 56 
 
 - 31.95 
 
 * 
 
 t + 4 
 +4 
 
 +0. 00014 
 
 -0. 0014 
 
 +0.0060 
 
 -0. 404 
 
 +1. 188 
 
 + 4.06 
 -11. 30 
 
 + 34.07 
 
 << 
 
 ( +40+34 
 0+40+34 
 
 +0.00005 
 
 -0.0005 
 
 +0. 0021 
 
 -0. 224 
 +6. 480 
 
 + 1.53 
 -47. 37 
 
 +120. 37 
 
 * 
 
 t+0+20+34 
 
 
 
 
 +0. 214 
 
 - 1.66 
 
 
 1 
 
 f+0+60+54 
 
 
 
 
 +5. 977 
 
 -36. 82 
 
 
 
 (0-0 ) cos 
 
 
 
 
 
 
 
 n 
 v 
 v 
 
 20+24 
 t-^+20+2J 
 - t+^+20+24 
 
 -0. 00188 
 
 +0. 0143 
 
 -0. 0489 
 
 -1. 141 
 
 +0. 235 
 +1.12 
 
 + 7.14 
 - 1.12 
 - 7.39 
 
 - 20.54 
 
 v 
 
 <!> 
 
 -0. 00059 
 
 +0. 0051 
 
 -0. 0189 
 
 -0. 357 
 
 + 2.62 
 
 -8.20 
 
 i 
 * 
 
 i +40+44 
 V&+40+44 
 
 +0. 00155 
 
 -0. 0141 
 
 +0. 0540 
 
 -0. 975 
 +0. 939 
 
 + 7.39 
 - 7.27 
 
 + 23. 43 
 
 > 
 
 *+^+20+24 
 
 
 
 
 -1.12 
 
 + 7.39 
 
 
 I. 
 
 20+ 4 
 e-^+20+ 4 
 - +^+20+ 4 
 
 +0. 00068 
 
 -0. 0058 
 
 +0. 0222 
 
 +0. 847 
 -0.17 
 -0. 828 
 
 - 5.79 
 + 0.93 
 + 5.96 
 
 + 18. 12 
 
 1 
 
 <l> +4 
 
 +0. 00021 
 
 -0. 0020 
 
 +0. 0085 
 
 +0. 265 
 
 - 2.10 
 
 + 7.15 
 
 t 
 
 +40+34 
 ^+40+34 
 
 -0. 00056 
 
 +0. 0056 
 
 -0. 0239 
 
 +0. 724 
 -0. 697 
 
 -5.90 
 + 5.80 
 
 - 20.35 
 
 1 
 
 *+0+20+34 
 
 
 
 
 +0. 828 
 
 - 5.90 
 
 
 
 
 
 m' 3 
 
 
 
 m' 3 
 
 
No. 3.] 
 
 MINOR PLANETS LEUSCHNER, GLANCY, LEVY. 
 
 95 
 
 TAM.B XIX. 
 W.' 
 
 Unit-l". 
 
 
 
 
 e* 
 
 
 
 ^ 
 
 
 
 
 r 
 
 
 
 IP* 
 
 w 
 
 * 
 
 
 
 
 
 K* 
 
 * 
 
 " 
 
 
 -#+ 
 
 
 
 
 +0. 2108 
 
 - 1.059 
 
 +2.379 
 
 
 
 
 ^++40+44 
 
 
 
 
 -0. 2108 
 
 + 1.059 
 
 -2. 379 
 
 
 
 y 
 
 l 
 
 
 
 
 +0.843 
 
 - 4.236 
 
 
 
 
 V 
 
 +40+44 
 
 
 
 
 -0.843 
 
 + 4.236 
 
 
 
 
 
 iJ+ 20 24 
 
 - 294.9 
 
 + 740.6 
 
 -733.9 
 
 -1.200 
 
 + 7. 772 
 
 
 
 
 
 _^_j_ t -j- 20+24 
 
 
 
 
 -0. 875 
 
 + 5.583 
 
 
 
 
 
 <^++ 20+24 
 
 + 294.9 
 
 - 740.6 
 
 +733.9 
 
 +0.274 
 
 - L697 
 
 
 
 
 
 ^++60+64 
 
 
 
 
 +1.800 
 
 -1L658 
 
 
 
 
 
 ^+2* 
 
 
 
 
 -0.105 
 
 + 0.529 
 
 
 
 
 
 +2+40+44 
 
 
 
 
 +0.105 
 
 - 0.529 
 
 
 
 
 Jjf 
 
 * + ^ 
 
 
 
 
 -0.227 
 
 + L344 
 
 
 
 
 if 
 
 e -j-40-f-3J 
 
 
 
 
 +0.227 
 
 - L344 
 
 
 
 
 if 
 
 ^+ e 20 4 
 
 
 
 
 +1.758 
 
 -10.233 
 
 
 
 
 if 
 
 t!i+ +20+ 4 
 
 
 
 
 +1.083 
 
 -6.493 
 
 
 
 
 tf 
 
 ^+ t+20+34 
 
 
 
 
 -0.204 
 
 + 1. 377 
 
 
 
 
 rf 
 
 ^+ +60+54 
 
 
 
 
 -2.637 
 
 +15.350 
 
 
 
 
 ,2 
 
 -#+ 
 
 + 384 
 
 -1410 
 
 
 
 
 
 
 
 s 
 
 _^+ t 4^44 
 
 +1679 
 
 -6656 
 
 
 
 
 
 
 
 
 +20+24 
 
 +1180 
 
 -2963 
 
 
 
 
 
 
 
 if 
 
 +20+24 
 
 -1180 
 
 +2963 
 
 
 
 
 
 
 
 if 
 
 y''+ 
 
 - 384 
 
 +1410 
 
 
 
 
 
 
 
 f 
 
 ^+ +40+44 
 
 -1679 
 
 +6656 
 
 
 
 
 
 
 
 J 
 
 -#+ t - 4 
 
 - 285 
 
 +1210 
 
 
 
 
 
 
 
 
 -^+ -40-34 
 
 -2460 
 
 +8138 
 
 
 
 
 
 
 
 B jn' 
 
 +20+ 4 
 
 - 318 
 
 +1081 
 
 
 
 
 
 
 
 If* 
 
 -20- 4 
 
 + 318 
 
 -1081 
 
 
 
 
 
 
 
 
 #+ + A 
 
 + 285 
 
 -1210 
 
 
 
 
 
 
 
 *>!' 
 
 (f>+ +40+34 
 
 +2460 
 
 -8138 
 
 
 
 
 
 
 
 
 (0 ) sin 
 
 
 
 
 
 
 
 
 
 1 
 
 -tf+ -20-2J 
 
 
 
 
 +0.549 
 
 - 3.40 
 
 
 +0.00090 
 
 -0.0068 
 
 5 
 
 J+ +20+24 
 
 
 
 
 -0.549 
 
 + i40 
 
 
 -0.00090 
 
 +0.0068 
 
 Tf 
 
 _^+ t_20- 4 
 
 
 
 
 -0.407 
 
 + 2.75 
 
 
 -0.00032 
 
 +0.0027 
 
 Jf 
 
 #+ +20+34 
 
 
 
 
 +0.407 
 
 - 2.75 
 
 
 +0.00032 
 
 -0.0027 
 
 
 
 
 m' 
 
 
 
 m" 
 
 
 m 
 
 /j 
 
96 
 
 MEMOIRS NATIONAL ACADEMY OF SCIENCES. 
 
 TABLE XX. 
 
 [Voi.xiv. 
 
 [(1-ecose) W 2 ] 
 
 Unit- 4th decimal of a radian. 
 
 0. 
 
 *. 
 
 - 
 
 - 
 
 - 
 
 w 
 
 w 
 
 wo 
 
 w 
 
 w' 
 
 , 
 
 w 
 
 
 
 
 
 
 +0. 01022 
 
 -0. 0513 
 
 +0. 115 
 
 
 
 9* 
 
 
 + 18.6 
 
 - 68 
 
 
 +0. 187 
 
 -1.76 
 
 
 +0. 00043 
 
 -0. 0039 
 
 9 " 
 
 
 
 
 
 +0. 296 
 
 -2.46 
 
 
 +0. 00020 
 
 -0. 0020 
 
 
 
 
 
 
 -0. 186 
 
 + 1.34 
 
 -4.8 
 
 
 
 ^ Tj' 
 
 j 
 
 - 13.8 
 
 + 59 
 
 
 -0. 529 
 
 +4.34 
 
 
 -0. 00075 
 
 +0. 0065 
 
 
 
 '20+24 
 
 - 14. 29 
 
 + 35.9 
 
 - 36 
 
 -0. 1006 
 
 +0. 647 
 
 -1.91 
 
 -0. 000055 
 
 +0. 00041 
 
 ^' 
 
 20+ 4 
 
 
 
 
 +0. 1377 
 
 -0. 811 
 
 +2.19 
 
 +0. 000020 
 
 -0. 00017 
 
 ]j2 
 
 40+44 
 
 + 81.4 
 
 - 323 
 
 
 +0. 477 
 
 -3.64 
 
 
 
 
 7 l' 
 
 40+34 
 
 - 119.2 
 
 + 395 
 
 
 -1.295 
 
 +9.36 
 
 
 
 
 5 /J 
 
 40+24 
 
 
 
 
 +0. 921 
 
 -6.09 
 
 
 
 
 ; 2 
 
 40+34 -I 
 
 
 
 
 +0. 036 
 
 -0.32 
 
 
 
 
 
 (0-0 )sin 
 
 
 
 
 
 
 
 
 
 , 
 
 20+24 
 
 
 
 
 -0. 0266 
 
 +0. 165 
 
 -0.49 
 
 -0. 000044 
 
 +0. 00033 
 
 n' 
 
 20+ 4 
 
 M.' 1 .il 
 
 ! 
 
 
 +0.0198 
 
 -0. 134 
 
 +0.43 
 
 +0. 000016 
 
 -0. 00013 
 
 ) ) 5 
 
 40+44 
 
 ,..!, 
 
 
 
 +0. 151 
 
 -1.16 
 
 
 +0. 00031 
 
 -0. 0027 
 
 i) r;' 
 
 40+34 
 
 
 
 
 -0. 334 
 
 +2.47 
 
 
 -0. 00052 
 
 +0. 0045 
 
 9" 
 
 40+24 
 
 
 
 
 +0. 165 
 
 -1.24 
 
 
 +0. 00015 
 
 -0. 0014 
 
 
 
 m' 
 
 m' 2 
 
 m' 3 
 
 
 + 
 
 
 (V'.XW 'I 
 
 i of ;. 
 
Ho.*.] 
 
 MINOR PLANETS LEUSCHNER, CLANCY, LEVY. 
 
 97 
 
 * 
 
 2 
 I 
 
 
 8 
 
 o 
 
 kO i < OO 
 
 d MOO 
 
 7? 
 
 
 X 
 
 \ 
 
 S 
 
 sr <M r-i 
 
 w . 
 
 >. 0i5 
 
 8 S OT 
 + i + 
 
 E 
 
 -1 
 
 9 
 
 1 
 
 
 i 
 
 C4 O 
 
 o o 
 
 I-H CO IO 
 
 lOO 
 
 CH SO rl 
 l-l 
 
 + 1 
 
 
 - 1 
 
 9 
 
 
 
 i-H r*- i-H <N 09 O 00 *O 
 CO CO C*J O ^ C^ <N N *& M 4 CO I s - Ol *T5 OS 
 OO< i O^^t-CCGO*O -^(NOC^C5O Ol^-t* 
 
 
 ? ; [ 
 
 
 s 
 
 OOOCOCOOOiOCio i-i-<o6t^OCi rH*-iO 
 
 + 1+ ++ < 7++ i ++7 i + +1 + 
 
 
 
 
 g 
 
 c^ o "oi ifl r^ co^'i N re o ^r -^ i ^* N o co 
 Swc^ic-f^rcseow csoo^C'^'-o OO^H-^* 
 O O r-l O N 00 d ">" ^< S 1O CO 00 COC0O4 
 
 t 
 
 
 1 
 
 
 o o" o 1-1 i -<r * d o o N o o IN o o o 
 J^+ 1 + 1 +.1^ 1 +11 + 1 1 + 1 
 
 I 
 
 
 
 
 <MO"'m O -J-'a'co'T So"lM 00 "O V- C - 
 
 lOr^co^^rt^r- lOsco csoiodoo Nt^oo 
 
 OQt^O30-^r~i-H t^in^lOOt^ Ot^C< 
 O O -N id O> TT O OOMt^OCC OOO 
 
 ' 
 * - ^ 
 
 "Sog 
 
 
 
 O O O O O O i-l i-i<D O O O O O O OOO 
 
 + i + ++ i ++ 1++.L -^ + i + 
 
 
 
 
 1 
 
 N 00 -^ 
 O Mlfl 
 O C*5 ^ 
 S o c 
 . 5 5 
 
 OO OO 
 
 + 1 + 
 
 
 35s 
 
 1 
 
 3 
 
 WOOC: ^"t^CJCO CSOOt^ CC O5t^QO 
 
 2;oco o^*r~c<i <35 '^'^~'55^ "^* t ^2* 
 
 t 
 
 ? 
 
 
 
 o'oo o' o o' o o o o' o' o o o' o o' 
 1 1 + 1 + + + + 1 1 +J_ 1 1 + 1 
 
 
 
 
 S 
 
 oo * N oo'ir'n^' t^ o a> ~O ^-'oc'o' 
 t^OO or^ccc^ COT^SSQ?'? . r^oo-^ 
 
 
 III 
 
 
 
 o'o'o o' o o' o' o" o o o o o o o o' 
 
 ++ ' . + -'.-l-l ' ++ ' + + 1 + 
 
 
 
 
 ? 
 
 * s c 
 
 ^^^^^,^4 i ^^^ i 
 
 N TCOC^CO CT? N 1M <I? 
 
 ++++++ 1 ++++ + 1 
 
 ^> ^ <d OS ^ ^ ^ ^> ^ ^ ^> ^ ^ 
 
 ciCM^rr-^ 1 ^- c^iC^^^ -^r -^ 
 
 
 a r^fr 
 
 
 
 x ^ v M C4 C 
 *=" B* *=* =- V S> V (B w V C>%> V f> V B- V B- 
 B* R B C- p- p. ^ fy. ff. ^ p. 
 '<^N 
 
 
 
 
 N 
 u 
 
 110379 22 7 
 
MEMOIRS, NATIONAL ACADEMY OF SCIENCES. 
 
 [Vol. XIV. 
 
 s 
 
 ! 
 
 
 
 && 
 
 ?? /- 
 
 
 CO 
 
 
 
 -, 
 
 1 ~ 
 
 
 
 CD 
 
 ' 
 
 
 ^-* " ' 
 
 S 
 
 
 
 
 
 
 
 
 
 S 
 
 g 
 
 
 
 3 
 
 CD 
 
 rH 
 
 
 
 
 ^ ; 
 
 
 8 . 
 
 
 
 :v'S 
 
 ivi? c," x r-t -! 
 
 *v C f-* C* C 1 "' 
 
 ' ""*" ^ dr 
 
 1 
 
 1 + 
 
 W 
 
 1 " u ^' 
 
 '-*, O '" "^ X 
 -i- ( -J..4- i *. 
 
 f , -f r 
 
 3 
 
 O COCO 
 ^ CO CO 
 
 CO CO W CO CM O 
 M 1 OS rH t>- CO i i 
 
 ooirio^ NN 
 
 ? 
 
 i7-,'T' 
 
 j,ij;- >_ X 
 
 r- 'v tc v i, .T; 
 
 (-' -^t ;c i"* !'* 
 
 r.i - c- o o 
 
 
 +. 1 .+. 
 
 + 1 1+ + I 
 
 
 
 
 I 
 
 -r. r i 
 
 COO Ci 
 rH CQ 1O 
 COITUS 
 
 IO rH C35 O ^ *O 
 rH OO TT CO U5 CO 
 O t^lN IO CON 
 
 S -J C* 
 
 ' 
 *5f*' c^ >-- .- 1 
 
 T^t - } ta .- , <jf '\~, 
 
 C C .^ <** . ru 
 
 I Bill 
 
 V ^^ ^i.- V 
 
 
 '' + J- 
 
 rH O> CO O' O' O 
 +++ 1 1 + 
 
 tr ^ 
 
 " *H- 
 
 ^ C C 1 C 1 ': 
 
 - 14 
 
 
 9 
 
 
 
 
 1 
 
 S 
 
 IO CO CM 
 O * CO 
 OOO 
 
 CO 
 
 +4- 1 
 
 O CD O CO O CD 
 <f "5<M <N T)< (N 
 
 CNO NO oo 
 o' o o o o' o 
 
 I++ 1 1 -f 
 
 
 S 
 
 
 S 
 
 SSS 
 
 o'o'o' 
 
 1 1 + 
 
 O rH t^ CD N rH 
 
 o' o o' o o o' 
 
 + 1 1+ + 1 
 
 1 
 
 c ~. 
 
 11 II1I 
 
 C- 0- C C 
 
 t. 4- -i ; 
 
 l|i 
 
 SS8 J < i 
 
 5ff 
 
 s 
 
 CM N CM 
 
 CN COCO 
 
 t-H OO 00 CO OO ** 
 !> ift CO CO rH CO 
 
 
 ^^ 1||1 
 
 J'j i? fj 
 
 r 5 if 
 
 
 o' o o' 
 
 OOOO OO 
 
 
 v w O O 
 
 
 
 H h 1 
 
 1 -f-f 1 1 + 
 
 
 ~ z> o o C 1 o 
 
 =-oc 
 
 
 
 
 
 i i ! f- 
 
 f 4~r 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 s 
 
 
 
 
 
 7 
 
 o 
 
 
 
 
 
 
 
 ^,^^, rj 
 
 o 
 
 
 E H 
 
 
 
 '"I* T" ^V 
 
 *^ CO U 
 
 
 !c' | 
 
 
 
 <BcTci J 
 
 a ""a 5 
 
 r" r' f 
 
 ^t- < ^PC^' 
 
 
 
 tit - 
 
 + + 4, 
 
 -f- - 
 
 4-+ ! -f4--t"H 
 
 
 
 
 S.-S.-3.-S- ^ -S.-S- 
 
 
 
 
 
 V 
 
 XV V 
 
 ff p* w 
 
 
 "' 
 ~* 
 
 -a^-iT 
 
 
 R- 
 
 f=* P* K- 
 
 
 
 
 , * 
 f 
 
 iS 
 
 ~ s 
 
 -SS 
 
No. 3.] 
 
 MINOR PLANETS LEUSCHNER, CLANCY, LEVY. 
 
 TABLE XXIIa. 
 
 99 
 
 
 P 
 
 ^ 
 
 
 
 
 
 
 
 > 
 
 . 
 
 - 
 
 
 #+20+24 
 
 - 0. 614 
 
 + 4.059 
 
 -10.3 
 
 
 
 ! 
 
 20+24 
 
 - 4.255 
 + 2. 791 
 
 +27.89 
 -23.39 
 
 
 - 271.5 
 + 167.4 
 
 + 636.6 
 - 637.4 
 
 $ 
 
 20+ 4 
 
 #+40+34 
 
 + 5.444 
 -4.558 
 
 -31. 91 
 +33.80 
 
 'V 
 
 i K 
 
 ' -3d) doiriw n 
 
 1 
 
 40+44 
 #+20+24 
 #+60+64 
 -#+20+24 
 
 + 0. 11 
 +14.90 
 
 
 n bn <fvil> 
 L>a$i sv/ IF/ 
 
 +1514 
 +1360 
 -1227 
 - 273 
 
 -5780 
 -3387 
 +6415 
 + 179 
 
 'J 
 irf 
 
 4 
 40+34 
 #+20+ 4 
 #+20+34 
 #+60+54 
 -#+20+ 4 
 
 + 0.13 
 -44.62 
 
 F 
 
 fj |, ^00 .,_ fl 
 
 I ni( l,; 
 
 291 
 
 +1974 
 - 222 
 
 +7160 
 +2452 
 + 536 
 -9002 
 +1012 
 
 ^* 
 7" 
 
 nr/7.x >I<JBT 
 
 40+24 
 
 - 0.06 
 +30.53 
 
 
 ro 4 I)j )> 
 
 ,!* n.d. 
 
 
 * '( 
 
 40+34-2" 
 (0-0 ) sin 
 
 + 0.34 
 
 ^i^I + <lf)< 
 
 S 60') t- 1) ([ 
 
 rj-W-w 
 
 . 6 
 
 | 
 
 20+24 
 #+40+44 
 
 - 1.64 
 + L014 
 + 0. 782 
 
 +10. 18 
 + 8.43 
 -5.96 
 
 c.Ej-.irx 
 
 fiO^ J)T 
 
 
 < 
 
 ^ 
 
 
 
 20+ 4 
 # +4 
 #+40+34 
 
 40+44 
 
 + L22 
 + 3.249 
 - 0.579 
 
 ' 
 
 + 7.81 
 
 - 8.26 
 -30.12 
 + 4.74 
 
 [(.'Wnr-.'WX 
 
 MO *-!)]- 
 
 [ !In dtldfi n 
 
 ' L'W ] ,[,TT ) 
 
 r V l.^ui/t 
 
 !$ 
 
 4 
 40+34 
 
 + 3.25 
 -16. 10 
 
 ]-'^-V*i: 
 
 <*n*-nj-i 
 
 f,^ ^6 , - 
 
 )fiw-H 
 
 * 
 
 40+24 
 (0-0 )' cos 
 
 + 7.64 
 
 if , 6 rs 
 i\t\ i "^^ 
 
 ,(,.- 
 
 1 , ;*lf 'Ms 
 
 eoo >- 1)4- 
 
 *{.>. lllftli- 
 
 # 
 
 d$n 
 
 - 0.356 
 + 0. 266 
 
 + 2. 62 
 -2.10 
 
 ^ moil li^loq 
 qolavsb orfi 
 
 >J ijaliniifc nfi: 
 
 *nul firfT 
 
 niii n nl 
 Jaup') r.irfj 1< 
 
 
 
 m" 
 
 .1 i tuS 
 
 m' 
 
 In the construction of Tables XXI and XXII it is necessary to compute 
 
 :io1- L] uvti' --tj ."(.ijfttu'}/!l(.-> nf) T>J] 
 
 . f n-> noiJaup" '.JT 
 
 
 f(r 2 - 
 
 as one factor of a product, but the more complete tabulation is best arranged as follows. This 
 function gives all of the terms of the first order in the mass in W t -[ FJ. Let 
 
 
100 MEMOIRS NATIONAL ACADEMY OF SCIENCES. [voi.xiv. 
 
 and denote first order terms in F 3 -[F 3 ] and TP 4 -[FJ by W 3 " and W 4 ", respectively. 
 Then because of the similarity in the equations for these functions of successive ranks, the 
 sum 
 
 W,"+ W S "+W 4 " 
 
 can be computed by Z 70, eqs. (117), (118), (119). The coefficients F, G, B are tabulated in 
 Tables XXIII, XXIV, XXV. The mass factor ra' is, of course, implicitly contained in the 
 
 tables. 
 
 it 
 Eliminating the distinction between </> and , the function is 
 
 W t "+W 3 "+W<" 
 
 in which the coefficients A p _ g , determined by Z 71, eq. (121), are tabulated in Table XXVI. 
 The coefficients A M in the function 
 
 (l-cos) (F 2 " + W 3 " + W 4 ") 
 
 are computed by Z 71, eq. (123) and are tabulated in Table XXVII 
 By means of Table XXVII we readily compute 
 
 [(1-ecosO (F/'+F s "+W 4 ")] 
 tabulated in Table XXVIII. 
 
 Proceeding now to the determination of 
 
 [(1 - e cos e) W3 
 
 (from which we shall subtract [(1 e cos) W 3 "], already included in Table XXVIll), we have 
 by Z 53, eq. (95) 
 
 in which all quantities are known. The integration gives W 3 [ TFJ. 
 
 Having computed W 3 [ W 3 ], [ W 3 ] can be obtained from Z 53, eq. (95). 
 
 The function [T t ], computed from Z 53, eq. (94), is tabulated in Table XXVUIa. 
 
 In a manner similar to the development of equations for W, and f W t ], the right-hand side 
 of this equation, when computed, can be segregated into portions independent of tf>, terms 
 multiplied by cos </>, and terms multiplied by sin 0. It is of the form 
 
 A + B cos ^ + C sin ^ 
 
 where A, B, C are too complicated to be written analytically, but can be written by inspection 
 after the computation has been performed. 
 
 The equation can then be written in the three following equivalent equations: 
 
 - 
 
 _ W ,A. 
 
 in which we define 
 
NO. s.i MINOR PLANETS LEUSCHNER, CLANCY, LEVY. 101 
 
 From the first two equations we compute 
 
 *i- *-*,-,(" (A-r)B)d0 
 
 Let J 
 
 3 = fc/J COS 4> + fcJ 8m #' 
 
 * cr T- Qt ' 
 
 a 
 
 Then from the second and the third equations 
 
 cos $ + (7 sin - (^, 
 
 - 
 
 -si" !' 
 
 
 By inspection of , the function [yj can be written, and itfj/J added to [zj-ij[yj gives [zj. 
 Finally, 
 
 [W*] = [zJ + [yJ cos + [sJ sin ^ 
 and 
 
 [(l-ecos)FJ 
 is readily computed from IT,, which is tabulated in Table XXVTII&. 
 
 But this function contains [(1 - e cos e) W 3 "], abeady included in Table XXVHI. By Z 69 
 
 Subtracting Table XXVUIc from [(l-ecos e) Wj we have 
 
 [(1-COS) (W.-W,")] 
 
 which is tabulated in Table 
 
 
102 
 
 MEMOIRS NATIONAL ACADEMY OF SCIENCES. 
 
 [Vol. XIV. 
 
 S 
 
 B 
 
 o 
 
 CM OS CO OS 
 
 rH 
 1 1 + + 
 
 CO 
 
 ^rl* O t US OS US t- 73* CO t CO CM OS CO 00 
 rH 33 !" O t-CMrHO t-OSrH CO ^ CO CO 
 rH CO US t-T|<t- COCM 
 rH CO CM 
 
 1 + 1 + 1 1 + + 1 ++ 
 
 ia 
 
 CO b- rH OSCM 
 CM 00 rH O CO 
 
 OS CM O 
 rH 
 
 + + 1 + 
 
 o> 
 
 TJ< t^CO CM 
 
 *d d H< CD 
 
 rH 1C CO 
 rH 
 
 1 1+ + 
 
 1* 
 
 t CM *O rH O CO *C Cn OS OS T}* -^ t~- 10 iO 
 rH ^ CO rH OS US CM rHrH 
 CM US rHCO 
 
 1 + 1 + 1 1 + + 1 ++ 
 
 
 
 ^ CO b* OS b CM CO CO ^ O CM CO OS ^* ^] 
 ^ rHO b-CO COcOb- rHOOOSOS rHOCC 
 rHCO CM 1 ^ rHCOO b* CO O O b-lO 1 ^ 
 
 + + 1 + + 1 + 1 ++1 + \ 4 
 
 00 
 
 S 
 
 OS CMOO CM 
 
 CM CO US 
 
 1 1+ + 
 
 CO 
 
 t- CMOUS rH US fj< OS CD CO 00 rH CM t- t- 
 CO COOSt- CM 1C rH CO rHCDCM t- CM OO CO 
 CM USCO rHCMt-US rH 1C * i-l 
 CO rH t rH ^ 
 
 1 + 1 + 1 1 + +1 1 ++ 
 
 t- 
 
 t^ rH rH COrH rHOSO 00 00 r-1 CO CO CO ^ 
 CO -^t- COt- OCOOS t^OrHCM OOCOO 
 rHOO COO CMCCCM t- rH f- (M t- CO r- 
 rH CM COOS OS <? CO rH CC 
 CM CO rH & 
 
 ++I ++I+ I++I +14 
 
 r* 
 
 CO 
 
 >d o co co 
 
 rH CO CO t - 
 1 1 + + 
 
 CO 
 
 fC COCOCO OrHCOCM CMrHCO lO CO rH rH 
 O COCOCO -^COOSt- COOSO OOSCOCO 
 ^ t* O rH U5 00 b- rH OS N rH rH rH rH 
 iO rH O rH CO 
 
 1 + 1 +11+ +1 1 ++ 
 
 ^tf OO b CO CO ^ O CM CO C5 OS C5 C*-^ CM 01 
 rH rHCO -^OS CMCCCM OOCOtliO OOOST 
 
 + +i + +1+ \++\ +7n 
 
 to 
 
 CO O> Tfi OS 
 CM COt- OS 
 
 co 
 
 OS COkOOS T^^*^OS OOOO CMlClClC 
 rH -<^ CO CO tCOCOO ^t" CM t-^ 1C CM OS OS 
 CO OOOS rHOSOCO rH-^CO rHCOrHrH 
 t- rH rH 10 (MOD 
 
 1 + 1 + 1 1 + + 1 ++ 
 
 CO 
 
 US OOUS ^*^ rHrHCO t^"t COt^- USOSt' 
 rH rHOO US O CM rH CO l^ O rH CM t^ O C* 
 
 CO rH rHC 
 
 + + 1 + + 1 + 1 ++7 + 1 4 
 
 
 8 rHCO , 
 
 >o os O OS 
 
 ^ -^ rH CM 
 
 >< 
 
 CO t--^"? OOl-^CM lOCDiO t-OOOSOS 
 JM ^ T CO iS -*f 00 t CO OS OS iH t- OO OO 
 
 CXI 
 
 US CMOS CS-*}< COUSO CMCOrHQ rHCOT 
 * t-CO O CM rHCMrH rH CM rH US CO CM If 
 
 M 
 
 1 1+ + 
 
 US CM rHCO CM CM 
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No. 8.] 
 
 MINOR PLANETS LEUSCHNER, GLANCY, LEVY. 
 
 103 
 
 
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 MEMOIRS NATIONAL ACADEMY OF SCIENCES. 
 
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No. 8.] 
 
 MINOR PLANETS LEUSCHNER, GLANCY, LEVY. 
 
 105 
 
 8: 
 
 + i + i ++ i + i + ++ i i 
 
 - .- 
 
 + 1 
 
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108 
 
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 1C CO M 
 1C t^. 
 
 CO CO COO 
 CO CO CM (M 
 
 rH 1C IN ^f 
 
 e 
 
 CM IN 
 
 rH OO 
 
 en t~ 
 
 00 CD 
 rH t- 
 
 
 
 IN 
 
 rH 
 
 IN CO 
 
 rH IO 
 
 rH t~ 
 
 
 
 1C 
 
 
 
 
 
 
 rH 
 
 rH 
 
 CO 
 
 
 
 rH 
 
 
 
 i 
 
 + + 
 
 + 1 
 
 1 1 + 
 
 1 1 + 1 
 
 1 + 1 
 
 1 ++ + 
 
 + 
 
 1 J^ 
 
 1 + 
 
 
 o 
 
 
 
 r-. 
 
 
 
 
 
 
 
 
 g 
 
 ss? 
 
 oo? 
 
 OOS 
 
 CO CO CO ^ 
 
 1 ~ -f 
 CD rH 
 
 b-O CO ^ 
 
 CO CX CO 1C 
 
 b 
 
 rH 
 
 oo r-- 
 
 O CO 
 rH 
 
 - 
 
 
 r-co 
 
 CO 
 
 rH ^ H 
 
 !M 00 CO 
 CO CO IN 
 
 
 rH iC CO CO 
 rH rH 
 
 CO 
 CO 
 
 CO CO 
 CO CO 
 CM 
 
 gj 
 
 rH 
 
 
 
 
 
 
 CO 
 
 rH 
 
 
 
 
 
 
 1 
 
 + 1 
 
 + 1 
 
 'ii 
 
 1 ' i 1 
 
 1 ++ 
 
 1 +++ 
 
 + 
 
 1 + 
 
 1 + 
 
 
 rH 
 
 5 
 
 ^ CO 
 1C CD 
 
 COCO 
 
 iC b- 
 
 o 
 
 O5 rH CO 
 
 co Ci b- 
 
 b--<J< (M 
 
 CD CO -^ CO 
 
 rH CO CO 1C 
 CO CD O 
 
 I-^ -^JH CO 
 
 OO CO CD 
 00 IM 
 
 rH CO O CO 
 
 sill 
 
 ia 
 b- 
 
 00 1C 
 OO CD 
 CISCO 
 CO t~ 
 
 00 1C 
 CO t^ 
 
 coo 
 
 rH 
 
 
 
 r-i 
 
 r-H 
 
 CO 
 
 rH CO 
 
 CM O 
 
 
 
 rH F 
 
 CO 
 
 
 l 
 
 + 1 
 
 ++ 
 
 l+ 
 
 +-1-1 
 
 1 + + 
 
 i ++ 1 
 
 + 
 
 1 + 
 
 1 1 
 
 
 
 CO 
 00 
 
 rH 
 
 -}H 1C 
 
 CO 
 
 Tf< CO 
 
 "1 
 
 CD CO 1C 
 1C CM 
 
 * 00 CO (N 
 -* 00 O f~ 
 (M CM 1C CO 
 CM 1OO CO 
 rH CM IM 
 
 (N 00 
 
 rH t-* f- 1C 
 
 b- O O rH 
 
 O CO CO CO 
 
 coo o ^ - 
 
 r- 1 CO rH 
 
 rH 
 
 CD b- 
 
 Sb- 
 co 
 
 1C 
 
 rH 
 
 rH CO 
 
 co en 
 
 cog 
 
 
 + 
 
 .1-1 
 
 1 + 
 
 x 
 
 + 111 
 
 1 + + 
 
 1 ++ 1 
 
 1 
 
 
 
 + 1 
 
 
 1C 
 CO 
 
 e 
 
 rHOO 
 
 O 
 coo 
 
 CD *^* 
 
 t-o 
 
 CO 
 
 OO d C5 
 
 coco co 
 
 CO t- CO 
 CO ^ CO 
 
 rp CO CO -^ 
 
 ^3 Tf* co t- 
 
 CD t~ CO 
 
 1C -^ J>- CO 
 OQ i i CO CO 
 
 ^ rH CO CO 
 
 1 
 
 rH 
 
 8$ 
 
 OC rH 
 * O 
 
 - C-l 
 SIC 
 CS 
 rH 1C 
 
 
 
 
 
 
 rH rH 
 
 IN IN 
 
 
 
 
 
 
 + 
 
 + 1 
 
 ++ 
 
 1 + 1 
 
 
 
 1 4- 1 I 
 
 1 
 
 1 + 
 
 1 1 
 
 
 8 
 
 
 
 
 
 
 
 
 
 
 O 
 
 i 
 
 CO CO 
 
 r t 
 
 CM CM 
 (M CM 
 
 isi 
 
 SCO COO 
 t- b- CO 
 
 5JSS5 
 
 CD CN CO 
 
 rH 00 rH 
 
 rH rH 1C lO 
 CO CO CD CD 
 IN IM 00 00 
 CO CO t- f~ 
 
 CO 
 
 CO 
 
 JO CD 
 rH rH 
 
 CD CD 
 CO CO 
 
 m in 
 
 
 + 
 
 1 1 
 
 + + 
 
 1 + 1 
 
 +77+ 
 
 + + + 
 
 MM 
 
 1 
 
 + + 
 
 1 1 
 
 
 
 
 
 
 
 
 6 eso 
 
 
 
 
 
 
 
 | -(- 
 
 
 
 
 iJLil, 
 
 
 
 
 
 
 
 
 
 rH rH rH rH 
 
 ++ 1 1 
 
 
 rH rH rH rH 
 + 1 1 + 
 
 
 
 
 
 
 
 
 rH rH 
 
 S g 
 
 g e s e 
 
 CN" CN~ 
 
 S S S 
 
 
 It"? 
 
 rH rH 
 
 
 
 1 1 
 
 + 1 
 
 
 MM 
 
 -|- | 
 
 1 1 1 1 
 
 ^^ 
 
 1 1 
 
 + 1 
 
 C 
 
 *" 
 
 . . 
 
 g S 
 
 "? 
 
 
 g g g 
 
 . . . . 
 
 g 
 
 
 g g 
 
 
 I 4 
 
 rH rH 
 
 + 1 
 
 1 1 
 
 + '.7 
 
 +7+7 
 
 1 1 1 
 
 + 1 + 1 
 
 1 
 
 +7 
 
 1 1 
 
 
 g 
 
 
 8 S 
 
 S $ S 
 
 g e^e.^ 
 
 S S S 
 
 s s s 
 
 g 
 
 S S 
 
 g g 
 
 
 
 
 o o 
 
 **~^T~ 
 
 **~^~G*~Q 
 
 z*~~~~~~ 
 
 N N M 
 
 c c o o 
 
 o 
 
 o o 
 
 - 
 
 
 " 
 
 ^ 
 
 O 
 
 -i^ 
 
 ^W 
 
 -K-r 
 
 O O O 
 
 
 
 ^ 
 
 C O 
 
 
 
 
 
 en 
 
 IOJCBJ 
 
 
 
 
 z .n aoci, 
 
 >M 
 
 ir V N- I- t .-* 
 
 . J <v- :-.- r-r 
 
 -ii i v i-r C- -t 
 
 & c : o V 
 
 t:g 
 
 
No. 3.] 
 
 MINOR PLANETS LEUSCHNER, GLANCY. LEVY. 
 
 TABLB XXVIII. 
 [(1 -e coe )( W t "+ W 3 "+ W t ")] Unit-4th decimal ot a radian. 
 
 
 
 Cos 
 
 
 
 W 
 
 w' 
 
 s 
 
 
 - 4. 1829 
 
 + 12.406 
 
 - 16.58 
 
 5' 
 
 
 - 68.61 
 
 + 347.9 
 
 
 V s 
 
 
 - 83.96 
 
 + 413. 1 
 
 
 f 
 
 
 + 83. 96 
 
 - 413.1 
 
 
 %9' 
 
 J 
 
 +134.0 
 
 - 714. 2 
 
 
 V 
 
 20+ 2J 
 20+ 4 
 
 + 70. 842 
 - 42. 107 
 
 [- 165.57] 
 + 147. 38 
 
 [+255.8] 
 -288.6 
 
 !V ! 
 
 40+ 44 
 40+ 3J 
 
 -345. 88 
 +876. 64 
 
 f+ 843.0] 
 [-1481. 7] 
 
 
 i? 
 
 40+ 2J 
 
 -514. 54 
 
 + 405.5 
 
 
 
 
 40+ 34 -.r 
 
 - 61.87 
 
 + 273. 1 
 
 
 ! f>! M i. 
 
 
 m' 
 
 , 
 
 
 TABLE XXVIIIa. 
 
 Unit- 4th decimal of a radian. 
 
 
 
 
 V 
 
 -t 
 
 
 
 i 
 
 I it T 
 
 
 
 
 
 
 
 
 Sin 
 
 
 
 
 
 
 
 
 
 w* 
 
 w 
 
 tc 
 
 w 
 
 
 
 A 
 
 
 
 
 
 
 ^+20+24 
 
 -0 
 
 00005 
 
 +0. 00073 
 
 -0.0682 
 
 +0.4056 
 
 - 1 TT .01 4 
 
 | ^#3 .fi ' ! S1I-) 
 
 .0 
 
 
 
 
 
 9 
 
 20+2J 
 
 0+ 
 
 
 
 -0. 3324 
 
 +2.1665 
 
 T! 
 
 ^ 
 
 
 
 
 +0. 3381 
 
 -2.5547 
 
 ' 
 
 
 
 
 
 +L0220 
 
 -7. 370 
 
 ?' ; SS+ 
 
 20+ J 
 
 .0- 
 
 
 
 +0.2654 
 
 -1.846 
 
 i)' 
 
 ^ +4 
 
 0+ 
 
 
 
 -0. 3622 
 
 +2. 472 
 
 '' 
 
 ^+40+34 
 
 0-i- 
 
 
 
 -1. 2106 
 
 +8. 472 
 
 
 
 
 m 
 
 
 m 
 
 /7 
 
 109 
 
 
 0/00 .0- 
 ;.f)0 I 
 
110 
 
 MEMOIRS NATIONAL ACADEMY OF SCIENCES. 
 TABLE XXVIII6. 
 
 [Vol. XIV. 
 
 Unit- 4th decimal of a radian. 
 
 
 Cos 
 
 ur 
 
 w-i 
 
 w 
 
 U) 
 
 UI 
 
 > 
 
 w 
 
 tc 
 
 u> 
 
 w 
 
 
 - e+<f> 
 
 -0.00004 
 
 +0. 00038 
 
 -0. 0032 
 
 -0.0005 
 
 -0. 4803 
 
 
 
 
 t +20+24 
 2t-^+20+24 
 <j>+2d+2J 
 
 0.00000 
 +0.00001 
 
 +0.00004 
 -0. 00023 
 
 +0. 0237 
 +0.0050 
 +0. 0726 
 
 -0. 15101 
 -0. 0318 
 -0. 4507 
 
 
 + 13. 16 
 - 0.81 
 
 - 30.86 
 + 1.31 
 
 
 2s +49+44 
 t+4>+40+44 
 
 +0.00004 
 
 -0. 00038 
 
 +0. 0153 
 +0. 0181 
 
 -0. 0770 
 -0. 0814 
 
 
 + 3.88 
 - 16.23 
 
 - 14. 38 
 + 61. 80 
 
 
 2t+<fr+60+64 
 
 
 
 -0. 0088 
 
 +0. 0576 
 
 
 - 3.0 
 
 + 15.7 
 
 7 
 9 
 IJ 
 
 y 
 
 20+24 
 t-4>+20+24 
 - t+<fr+20+24 
 
 1 
 
 
 .!! il ( / / 
 
 +0. 5242 
 +0. 1384 
 -0. 0508 
 
 +0. 0749 
 
 -3. 3539 
 -0. 7747 
 +0. 4660 
 
 -0. 2385 
 
 
 + 13. 16 
 + 14. 86 
 + 58.25 
 
 - 30. 86 
 - 11. 30 
 - 170.9 
 
 
 5 
 5 
 
 . +40+44 
 ^+40+44 
 
 
 
 +0. 1723 
 -0. 6378 
 
 -0. 8380 
 +4. 082 
 
 
 -154.6 
 - 16. 23 
 
 + 589.2 
 + 6L80 
 
 I? 
 
 t+^+20+24 
 
 
 
 -0. 1801 
 
 +1. 1267 
 
 
 - 7.71 
 
 - 6.66 
 
 3 
 
 e +^+60+6J 
 
 
 
 -0. 3275 
 
 +2. 032 
 
 
 +178.4 
 
 - 933.0 
 
 J 
 
 29+ 4 
 ,-^+20+ 4 
 - +4>+20+ 4 
 
 
 r 
 
 -0. 6099 
 -0. 1412 
 +0. 1220 
 
 +3. 634 
 +0. 6843 
 -0. 8554 
 
 
 + 10. 77 
 - 31. 34 
 
 - 49.04 
 + 118. 9 
 
 I/ 
 
 * + I 
 
 
 
 +0. 0524 
 
 -0. 4182 
 
 
 
 
 ? 
 
 , +40+34 
 Vi+40+34 
 
 
 
 -0. 0411 
 +0. 7660 
 
 +0. 2314 
 -5.430 
 
 
 +221.0 
 
 - 694.3 
 
 9' 
 
 +(/>+20+34 
 
 
 
 +0.0460 
 
 -0. 3060 
 
 
 + 14.12 
 
 - 26. 01 
 
 i' 
 
 f+^+60+5J 
 
 
 
 +0. 3718 
 
 -2. 0745 
 
 
 -287. 1 
 
 +1309. 3 
 
 
 (0-0 ) sin 
 
 
 
 
 
 _, _. 
 
 
 
 5 
 v 
 q 
 
 20+24 
 t-^+2(?+24 
 - t+^+20+2J 
 
 
 
 +0. 0490 
 +0. 0144 
 -0. 0542 
 
 -0. 2949 
 -0. 0705 
 +0.3582 
 
 
 
 
 5 
 
 t 
 
 
 
 +0. 5810 
 
 -4. 7017 
 
 
 
 
 5 
 5 
 
 t +40+44 
 ^+40+44 
 
 
 
 -0. 0618 
 -0.0453 
 
 +0. 4650 
 +0. 3389 
 
 
 
 
 >? 
 
 +^+20+24 
 
 
 
 -0. 0010 
 
 +0. 0290 
 
 
 
 
 1 
 
 20+ 4 
 t-^+20+ 4 
 - t+^+20+ 4 
 
 
 
 -0. 0364 
 -0. 0107 
 +0. 0402 
 
 +0. 2398 
 +0. 0585 
 -0. 2889 
 
 
 
 
 * 
 
 ^ +4 
 
 
 
 -0. 7670 
 
 +5. 4890 
 
 
 
 
 3: 
 
 t +40+34 
 ^+40+34 
 
 . 
 
 
 +0. 0459 
 +0. 0336 
 
 -0. 3715 
 -0. 2709 
 
 
 
 
 ^ 
 
 t+^+20+34 
 
 
 
 +0.0007 
 
 -0. 0220 
 
 
 
 
 
 m' 3 
 
 m' 2 
 
 m' 
 
No. 3.] 
 
 MINOR PLANETS LEUSCHNER, CLANCY, LEVY. 
 
 TABLE XXVIIIc. 
 [(l-coet)Tr"] 
 
 111 
 
 Unit- 4th decimal of a radian. 
 
 
 Cos 
 
 10 
 
 Ml 
 
 va 
 
 V 
 
 20+2J 
 26+ J 
 
 +60.76 
 -20.57 
 
 -152. 6 
 + 69.8 
 
 
 m' 
 
 TABLE XXIX. 
 
 T7nit-4th decimal of a radian. 
 
 
 Cos 
 
 te- 
 
 -! 
 
 w 
 
 * 
 
 w 
 
 < 
 
 1C 
 
 vfi 
 
 to 
 
 V 
 
 20+2J 
 20+ A 
 
 -0.00004 
 
 +0.00038 
 
 -0.0032 
 +0.5106 
 -0.6292 
 
 -0.0005 
 -3.0290 
 +3.463 
 
 +13.16 
 
 -30.9 
 
 
 (8-O t ) sin 
 
 
 
 r ~~ e ' 
 
 
 
 
 V 
 
 20+2J 
 20+ J 
 
 
 
 +0. 0092 
 -0. 0069 
 
 -0. 0072 
 +0.0094 
 
 
 
 
 S MS 3 - 1 ~ -e J 
 
 " l/ * 
 
 "* "'": 
 
 m' 
 
 These developments cover the function 17 within the extent of our tables. This does not 
 mean that W is always inclusive of all these terms, but that these terms occur in one or more of 
 the tables. With the exception of [(1 e cos e) W], which contains W 3 W t ", W is to be under- 
 stood to mean F== Fj + W ^ + [ jpj + ( ^// + Wy + Wf ") 
 
 W= W,+ W t ' + [W3 + (W,"+ W t "+ F 4 "). 
 
 The ascending powers of w, TJ, 17', f are selected independently in each function. 
 
 To avoid along series which is analogous in construction to T 2 , the function W t " + W t " + TP 4 " 
 is not tabulated. The sum of this function and Tables XVII, XVHI, XIX, XXIIa gives W. 
 Since W is so long and we only need W, it is not tabulated. The function 
 
 W= TPj_, 
 is given in Table XXIXa. 
 
 It is convenient to collect here [(1 e cos e) W], which is required later. The function is 
 given by the sum of Tables XVI, XX, XXI, XXVIII, and XXIX, and is tabulated in Table 
 XXIX6. 
 
 We shall also need the function 
 
 Evidently S can be written by inspection if Wis tabulated. If the double headings are retained in 
 the construction of H the mass factors and ranks are explicit as in the construction of W. If W 
 is not given, we can write by inspection E, (previously required in the computation), E 2 ' and 
 [EJ from F u F 2 ', and [FJ, respectively. The remainder, namely, ,"+"+/', can 
 be written from W 2 " + W s " + W 4 ", i. e., by inspection of Tables XXIII, XXIV, XXV. The 
 
 function 5- E is given in Table XXIXc. 
 
112 
 
 MEMOIRS NATIONAL ACADEMY OF SCIENCES. 
 
 [Vol. XIV. 
 
 ^ 
 
 
 
 
 
 HI aojl 
 
 
 
 CO CO 
 
 
 
 
 
 
 CO CO CO 
 
 t".' 
 
 ,..*,-. > -i..j 
 
 
 
 3 
 
 -f- 1 1 
 
 ' Ia ! ** Ml l!i: " 
 
 i : I 
 
 
 
 
 OS OS OS 
 iO iC iO f i i O CO 
 
 OOO CMt^cor-co 
 
 CO CO CO CO CO 
 
 
 
 
 M 
 
 3 
 
 r-t ^ rH i I CO CM rH rH 
 
 CO" f-i 3* rH lO 
 
 pasn M3M, mnnjoo aiqj up sraiai aaxSap paooas XX 
 
 
 
 
 1 ++ + 1 1 1 1 
 
 I++++ 
 
 
 
 
 
 sis s?? ss 
 
 U9t- S t~ 
 
 
 
 
 
 CM CO CM CO I 1 -- iO CM CO 
 
 CM OJO WCO 
 
 f CO * 1C CO CM K5 00 O OOO O rH VC rH COCOOO OO CO CO CM 
 
 
 
 S 
 
 o o o" o* o* co o" t-t 
 
 oo" o' us' O CM' 
 
 O: o O' ^ CM O* rH rH 1C 1C' CO -^ t-^ l.O O* ^' rH ^' CO Q 1C CM 
 COCOrH rHrHt^COrHCM CO OrH 
 
 
 
 
 +11 1 ++++ 
 
 + 1 1 1 1 
 
 ++I l+l 1 1 1 I+++I++I+ ++I 1 
 
 
 
 
 CO 
 
 00 
 
 i i 
 
 rH 
 
 i^ t~ CM O C*l OO OS t"- rH \fi CO CM CO CM GJ> 
 
 
 
 1 
 
 o* 
 
 O 
 
 O' rH* ^* rH CM* O* O' id rH CO CO O* O" CO rH 
 
 1 1 ++ 1 ++ 1 1 1 + 1 1 ++ 
 
 
 
 3 
 
 CM 00 
 
 CM lO OS* bU3rHCOO 
 ^ i t O CM ^* r-4 lO CO 
 CO ^f CM rH CD OS O 
 
 yoj<^. .0-r 
 
 1C O t^ CC IO 
 
 1 t * 
 
 
 
 
 CM CO 
 
 CO 
 
 
 
 
 
 1 1 + +++ 1 1 
 
 1 1 1 1 + 
 
 
 
 
 
 
 
 t--f-^;; , 
 
 
 
 
 
 OS 10 
 
 
 
 
 
 S 
 
 id rH CO* lO rH CO OS CM 
 
 CM lO CO t 1 " OS CM ^ i-H 
 H pH ^ * 
 
 rH 
 
 ?|S|S| 
 
 rH rH 
 
 CM rH O rH iO rH CO CO lO ^* rH rH CD CM *& 
 
 ______) 
 
 T 
 
 
 ++ 1 +111+ 
 
 + + + + 1 
 
 + 1 ++ 1 1 + 1 1 + 1 1 1 + 1 1 + + 1 ++ 
 
 rfT 
 
 S 
 
 
 
 
 
 i,J ilU'Hll 
 
 ... , 
 
 
 00 
 
 i<) i /i >ulv; 
 
 ,pM (i y> 4-- I '] 1<> noiiqo'M.'i iii) J.ti// . 
 
 'da) oiii 
 
 
 
 <NCOt- rH 
 
 
 
 
 
 3 
 
 CD* CO CM CD* CO ^* O CM 
 OS rH CO CO t** rH, t~ 
 
 CM CO OS r rH 
 
 r-* CO lO t^ rH 
 CM r-i O rH CM 
 
 co o co co en co co co co co oo CM rH oo t- oo rH a> CMCM^CO 
 
 lOCOOOCOOCCOrHCM CTSCMCMCTiCMGOSOCnCJlcN COCOCOO 
 r-COCMCO-'t 4 COCO CMi-HOrHOOCOt^t^CO t iOCOCM 
 OCO CM 1 ^ CMt^CMCMCOeft CMCM 1 ^ rHCOCMrH 
 rH CM rHrH CO rH T(< 
 
 lj iiu}:-. 
 
 hnjj 
 
 
 
 11+ 1 +++ 1 
 
 + 111 + 
 
 1 + 1 1 ++ 1 + + 1 + 1 + 1 1 ++ 1 1 + 1 1 
 
 
 ' ''/ 
 
 
 OS 
 
 
 
 
 M 
 
 
 00 CO O 
 
 CM 00 
 
 
 
 
 S 
 
 Tp OS OS 
 
 CM rH 00 
 
 rH 
 
 + + t 'r :i 
 
 2 S 
 
 CO <N 
 rH 
 
 l + 
 
 CO OS OS CO CD CM ^* CO O CO CO CO 
 
 "+++ +7+T+ 1 1 + 
 
 1 'J'i!l'". 
 
 , 
 
 wiT 
 
 i*f !>v.iu >- 
 
 I'v'-iv; : " 
 
 n 11 my\ ,- I)] ;mii )'J')Ho3 O-t ^lIOli!'* . ii- i 
 
 ;'(')'/i^J r*t 
 
 (i h- 
 
 >J*;li 
 
 
 
 ^X/ .\/7. ~A7. .Vf'A aoWuT 1<> >m;^ 
 
 
 
 
 
 
 
 v'/l/.X 
 
 
 ! 
 
 ^ ^ ^ ^, ^ '^ 
 
 ^^^^^ 
 
 ^^,^^, ^,^ ^,^,^,^,^^^^, T ,^ I ^n-n 
 
 
 
 
 CM * O4 Tji CM CO 
 
 CO CO tO 
 
 TCMCOCM Tf 00 CO CO lO CO >O t~ CM CM -d 
 
 
 
 
 ^ ^ ^i ^i ^ ^ 
 
 ce> co co co 
 
 ^ ^ ^ ^5 ^ ^S ^ ^ ^ ^ *^ ^) ^S ^S ^ ^S ^ 
 
 
 
 
 + ^. 4. + + 
 
 CM ^C^CO 
 
 TTCMCOCM -^00 -^C-JfMCOCM ^^J*OO -VCMCO 
 
 
 
 
 C, 
 
 " "(NCM 
 
 I'fXO ') 
 
 CM <M CM CM CM CM CM 
 1 1 
 
 
 
 
 
 
 jotjeui 'Jfl - 1 - ".";! nt. 
 
 
 
 
 B- 
 
 V 
 
 v 
 
 "s- i ,+>/> 
 
 
No. 3.] 
 
 MINOR PLANETS LEUSCHNER, GLANCY, LEVY 
 
 noiptujsuoo aq; nj 
 
 o >o eo i-i e> ! n o 
 + +1 I + 1+ + I 
 
 3 
 v 
 i 
 
 oo 
 
 + 1 
 
 M - 
 
 01 -f 
 
 k 
 
 H -f 
 
 ^ ~- 
 
 o *~^ o ^^ *o C"4 25 10 o ^ ^ c 
 
 rt s-i o ^o oo c^ i < o 35 -^ 
 
 < C^ <-^ ^ f^ fH 
 
 ii+7 i + i + i i + 
 
 Oil- CSJ 4- 
 
 ~J 
 
 
 ~ 
 
 COWIM 00 
 I l+l 
 
 ++ 
 
 
 
 110379 22 8 
 
 
 
 ' 
 
 
 
 
 ^-i J 
 
 * f > 
 
 
 1 
 1- 
 
 
 
 113 
 
114 
 
 MEMOIKS NATIONAL ACADEMY OF SCIENCES. 
 
 [Vol. XIV. 
 
 TABLE XXIXa Continued. 
 W. 
 
 fnit-l". 
 
 
 Cos 
 
 !0-l 
 
 ur- 
 
 w 
 
 w 
 
 w* 
 
 w<> 
 
 w 
 
 vfl 
 
 3 
 
 +40+44 
 - +40+44 
 +80+84 
 
 + 2549 
 - 3089 
 -11300 
 
 + 2164 
 + 8155 
 + 76250 
 
 
 -11.9 
 + 3.9 
 -8.9 
 
 1 
 + 70 
 
 (lOUOjJ/l ] 
 
 W 
 
 +40+54 
 +40+34 
 - +40+34 
 f+80+74 
 
 -11449 
 - 2661 
 + 6865 
 +50005 
 
 + 42212 
 - 27530 
 - 4540 
 -304611 
 
 
 + 1.9 
 +36.4 
 -20.3 
 +83.8 
 
 - 23 
 -241 
 +118 
 -248 
 
 
 -M" 
 
 +40+44 
 +40+24 
 - +40+24 
 +80+64 
 
 +26091 
 - 1356 
 - 2204 
 -73583 
 
 - 71730 
 + 30293 
 - 20846 
 +400009 
 
 
 -10.1 
 -25.5 
 +28.0 
 -41.9 
 
 + 83 
 +153 
 -153 
 +284 
 
 
 1* 
 
 f+40+34 
 - +4(9+ A 
 +80+54 
 
 -13756 
 - 3317 
 +36006 
 
 + 22165 
 + 18452 
 -172164 
 
 
 +10.1 
 -12.4 
 +16.6 
 
 - 65 
 + 64 
 -104 
 
 
 fr, 
 
 t+40+34-. 
 - +40+34 -JT 
 f +80+74-2" 
 +40+44 
 
 - 2011 
 + 1808 
 - 2381 
 +14204 
 
 + 14604 
 - 13617 
 + 18919 
 - 88026 
 
 
 - 1.9 
 + 1.9 
 - 1.1 
 
 + 5.7 
 
 + 14 
 - 14 
 
 + 10 
 
 - 42 
 
 
 ? -' 
 
 f +40+44 -.T 
 - c+48+24-S 
 +80+64 -.F 
 +40+34 
 
 - 554 
 - 3545 
 + 3827 
 -17503 
 
 + 140 
 + 22886 
 - 27870 
 + 99584 
 
 
 + 0.5 
 - 1.8 
 + 1.3 
 -3.7 
 
 - 4 
 + 14 
 - 11 
 + 28 
 
 
 
 (0-0 ) sin 
 
 
 
 
 
 
 - r-. -; o. g 
 
 n 
 
 20+24 
 
 +45+44 
 2f+20+24 
 
 + 767. 7 
 
 - 2820. 9 
 
 + 5210 
 
 + 1.265 
 
 r * ? 2. 19 
 
 - 5.34 
 + 0.78 
 - 0.55 
 
 +13.6 
 +22.7 
 - 6.0 
 + 3.4 
 
 1 
 
 26+ A 
 + ^ 
 +40+34 
 2 +20+34 
 
 - 570. 
 
 + 2421. 1 
 
 - 4950 
 
 - 0.455 
 
 + 1.63 
 + 5.94 
 - 0.58 
 + 0.41 
 
 -11.0 
 -37.3 
 + 4.8 
 -2.8 
 
 f 
 
 40+44 
 +20+24 
 - +20+24 
 2 +40+44 
 
 
 
 
 
 + 10.93 
 - 2.19 
 - 1.92 
 + 3.12 
 
 
 iV 
 
 ? 
 
 4 
 40+34 
 
 E 
 
 - 570. 
 + 6624 
 
 + 2421. 1 
 - 47448 
 
 - 4950 
 
 - 0.455 
 
 +23.8 
 
 + 5.94 
 - 23.00 
 -221. 9 
 
 - 7.2 
 
 ,y 
 
 + 4 
 
 - + 4 
 
 -18540 
 + 8414 
 
 + 123024 
 - 57880 
 
 
 -73.4 
 +36.0 
 
 +572. 4 
 282. 2 
 
 
 11" 
 
 + 24 
 
 +25564 
 +10478 
 
 -157424 
 - 70250 
 
 
 +87.3 
 
 +55.2 
 
 -652. 8 
 -374. 8 
 
 
 ," 
 
 <+ 4 
 
 -15678 
 
 + 94846 
 
 
 -69.9 
 
 +438.6 
 
 
 rt 
 
 '+ 4+^ 
 
 -25564 
 +22012 
 
 +157424 
 -121258 
 
 -511232 
 +359162 
 
 -23.1 
 + 9.9 
 
 +165. 
 - 77.0 
 
 
 f 1 
 
 + 4 
 + -T 
 
 +23524 
 -12048 
 
 -150306 
 + 76364 
 
 +498328 
 -261640 
 
 +14.8 
 -5.2 
 
 -112.0 
 
 + 45.8 
 
 
 
 (0-0 ) J coe 
 
 
 
 
 
 
 
 \> 
 
 t 
 
 + 4 
 
 
 
 
 
 - 0. 356 
 + 0. 26G 
 
 + 2. 623 
 - 2.100 
 
 
 m' 
 
 m' 2 
 
No. 8.] 
 
 MINOR PLANETS LEUSCHNER, CLANCY, LEVY. 
 
 115 
 
 TABLB XX IXo Continued. 
 W. 
 
 
 
 as s 
 
 I 1 
 
 Unlt-1" 
 
 
 Cos 
 
 w* 
 
 W 
 
 w> 
 
 - 
 
 I 
 
 [+ 6+ J 
 
 - 293. 4 
 
 + 913. 5 
 
 - 1400.1 
 
 
 . 
 
 t+30+3J 
 
 + 338.1 
 
 - 2315 
 
 + 9277 
 
 
 , 
 
 i+56+bJ 
 
 + 42.9 
 
 - 284.3 
 
 + 948.2 
 
 
 
 1+78+7J 
 
 + 10.5 
 
 79.2 
 
 + 288.5 
 
 5 
 
 $+30+34 
 
 + 6172. 8 
 
 - 20580 
 
 + 86549 
 
 
 -*+ 0+ A 
 
 + 511. 2 
 
 - 2834 
 
 + 7746 
 
 
 *+ 0+ ^ 
 
 - 467. 9 
 
 + 2335 
 
 - 6259 
 
 
 it+50+54 
 
 - 2217. 1 
 
 + 23971 
 
 -157308 
 
 
 fc+30+34 
 
 5.8 
 
 + 539 
 
 - 3713 
 
 S ^~ "r 
 
 |t+70+74 
 
 e!j- 364.3 
 
 + 3259 
 
 - 15083 
 
 J 
 
 $+30+24 
 
 - 8375.5 
 
 + 20591 
 
 - 95913 
 
 
 -fctJ 
 
 - 1023.4 
 
 + 4443 
 
 - 10251 
 
 
 |+ 0+24 
 
 - 92.3 
 
 - 444 
 
 + 3212 
 
 
 |+50+44 
 
 + 3383.4 
 
 - 34097 
 
 +214736 
 
 
 |+30+44 
 
 - 138.6 
 
 + 608 
 
 - 1089 
 
 
 $+70+64 
 
 + 583.3 
 
 - 4805 
 
 + 20748 
 
 f 
 
 i 
 
 <+ 0+ A 
 
 - 5022 
 
 + 24269 
 
 
 
 l 
 
 +50+54 
 
 -31492 
 
 +154465 
 
 
 
 
 
 +30+34 
 
 + 8169 
 
 - 18309 
 
 
 
 
 t+30+34 
 
 - 59 
 
 + 7449 
 
 
 
 . 
 
 +70+74 
 
 +12392 
 
 -182737 
 
 
 
 
 
 + 0+ 4 
 
 + 1133 
 
 - 5174 
 
 
 
 |+90+94 
 
 + 2342 
 
 - 25879 
 
 
 *?' 
 
 
 t+ 
 
 + 6153 
 
 - 26311 
 
 
 
 
 + 0+24 
 
 + 988 
 
 - 15732 
 
 
 
 
 +50+44 
 
 +88784 
 
 -357566 
 
 
 L5 
 
 _. 
 
 +30+24 
 
 -14498 
 
 - 3083 
 
 1 2 X 
 
 
 
 +30+24 
 
 - 1309 
 
 5 
 
 I& JK 'JC 
 
 
 
 +30+44 
 
 + 4878 
 
 - 31947 
 
 
 
 
 t+70+64 
 
 -37540 
 
 +626187 
 
 
 
 : 
 
 + 
 
 - 3487 
 
 + 12764 
 
 
 
 -1 
 
 f+90+84 
 
 - 7382 
 
 + 77025 
 
 
 r 
 
 ; 
 
 \t+ 0+ 4 
 
 - 5966 
 
 + 27801 
 
 
 
 1 
 
 tf+50+34 
 
 -61877 
 
 +192684 
 
 
 
 \ 
 
 t +30+ 4 
 
 - 1709 
 
 + 26144 
 
 1 I 
 
 
 I 
 
 t- 0+ 4 
 
 + 1693 
 
 - 6306 
 
 
 
 f+30+34 
 
 - 5297 
 
 + 28649 
 
 
 
 $+70+54 
 
 +28418 
 
 -377278 
 
 
 f 
 
 
 + 0+ 4 
 
 + 6846 
 
 - 30542 
 
 
 
 
 +50+4J-J 
 
 - 3191 
 
 + 15590 
 
 
 
 1 
 
 tf+30+24-JE 1 
 
 - 806 
 
 + 10210 
 
 
 
 i 
 
 ,+30+34 
 
 - 3829 
 
 + 33852 
 
 
 
 ; 
 
 ,+70+64-2" 
 
 + 932 
 
 - 14562 
 
 
 
 ~' 
 
 ,+ -2 1 
 
 + 1762 
 
 - 6460 
 
 
 
 
 mf 
 
 X 
 
116 
 
 MEMOIKS NATIONAL ACADEMY OF SCIENCES. 
 
 [Vol. XIV. 
 
 1 
 
 
 IO 00 IO O) 
 
 O 
 
 
 
 
 
 
 
 1 
 
 r. 
 
 CO ^f* CO CO rH 
 rH C6 00 INrH 
 
 8S 
 
 
 
 * ' 
 
 
 
 
 
 
 rH CM rH rH 
 
 rH rH 
 
 
 
 
 
 
 
 -3 
 
 
 1 4- 1 + + 
 
 1 1 
 
 
 
 
 
 
 
 1 
 
 
 " O rH -"if 
 
 M- 
 
 rH kO 
 
 rH 
 
 
 
 L" ' \tf i- 
 
 
 
 
 C! r-J 01 OS CM 
 
 jrfo 
 
 COCO 
 
 COO* 
 
 
 
 
 
 *3 
 
 S 
 
 gq iS 
 
 CO OS 
 
 * CO 
 
 *? rH 
 
 
 
 
 
 & 
 
 
 4-4-4-4-1 
 
 1 4- 
 
 4-1 
 
 1 4- 
 
 
 
 1 
 
 
 
 
 IN 
 
 O IN CO O CO 
 
 CMOS 
 
 05 10 
 
 tO 
 
 rH CO 
 
 
 
 
 M 
 OS 
 CO 
 
 O 
 
 
 g 
 
 *4* CM Os O CO 
 
 CM co 
 
 COOS 
 
 g>0 
 
 coo 
 
 CO OS 
 
 
 rH rH 
 
 00 
 
 
 
 CM 
 
 rH IO 
 
 rH 
 
 
 TC CM 
 
 
 
 
 
 
 
 
 
 
 CM 
 
 rH CO 
 
 rH rH ^ rH rH rH 
 
 
 
 
 1 14-14- 
 
 4-1 
 
 1 4- 
 
 4-4- 
 
 1 1 
 
 4- 1 4- 
 
 1 1 1 4-4- 1 4- 1 14-4- 14- 
 
 + 
 
 
 
 
 
 
 
 
 
 O> T|- 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 08^ COCM 
 
 COIN 
 00 OS 
 
 BS 
 
 fl< OS 
 CO CM 
 
 CM 
 
 rH 10 
 
 COCO t~ O * 1* t~ CO S 
 
 
 
 a 
 
 O IO CO CM rH 
 ^ rH CO 
 
 rH 
 
 *H 
 
 CM ^ 
 
 t-ilO 
 rH 
 
 00 00 
 
 COCO OS 
 
 rH O OO 
 CO rH 00 
 
 t-lOO 00p CO OS CO IM t^ t- O O 
 
 COiMt- CO 1 '? (MOOr-t CO OO 
 (Mrtr rHCO rHCOrH CO rH 
 
 co 
 
 
 
 
 
 
 
 
 rH 
 
 CM rH 
 
 
 
 
 14-111 
 
 ++ 
 
 1 1 
 
 4-1 
 
 4-4- 
 
 1 4- 1 
 
 4-14- II 4-14-4-M 4- I 
 
 1 
 
 
 
 
 
 
 
 
 
 CO iO 
 
 
 
 
 rH CO rf 
 rH t- O CM OS 
 O CX ^ * CM 
 
 r-H CO 
 
 COrH 
 
 SS 
 
 rHO 
 
 gg 
 
 CO OS 
 
 Tf CO CM 
 
 t^-tO rHrHCOCO OSOS OOO 
 0000t~ CMCM IM CM OO rH O 
 
 1 
 
 
 a 
 
 O O O OrH 
 
 O CO 
 
 OCM 
 
 00 
 
 rH rH 
 
 ^ O iO 
 
 CM rH ^< rHrH O O OO O O 
 
 o 
 
 
 
 4-14-4-4- 
 
 1 1 
 
 4-4- 
 
 1 + 
 
 1 1 
 
 4-4-4- 
 
 1 1 1 4-4- 1 1 4-4- 1 4- 
 
 + 
 
 -=>' 1C 1 
 
 x i* 
 
 R ? 
 
 j 
 
 J* rH t~ 
 
 O O O O O 
 
 to co 
 
 ^ CO 
 00 
 
 osco 
 
 co 
 
 (N IN 
 
 t- << 
 
 CO ~ CO 
 
 t-ooo 
 
 IN CO 
 
 CO iO 
 IO CO O O C^ CM rH ^4 rH O 
 
 IO CM CO CM CM O O O O O O 
 
 S 
 
 CO 
 
 s 
 
 M 
 
 a 
 
 ooo oo 
 14-1 II 
 
 oo 
 
 oo 
 
 1 1 
 
 o 
 
 OO 
 
 ooo 
 
 1 1 1 
 
 OOO OO O O OO O O 
 
 4-4-4- 114-4- II 4-1 
 
 o 
 
 1 
 
 ' 
 
 
 
 
 
 
 
 
 
 
 < c> 
 
 
 
 
 
 
 
 
 
 
 
 
 IO IN rH rH 
 
 IO CO 
 
 (M t^ 
 
 CO 
 
 
 
 CO CO 
 
 1 *~ CO 
 
 s 
 
 
 
 
 
 * IM 
 
 
 IN 00 
 
 co os 
 
 Tj OS rH O 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 S S 88 
 
 
 88 
 
 S 
 
 8S 
 
 888 
 
 . . t ft*' 
 
 s 
 
 
 
 o o oo 
 
 oo 
 
 o o 
 
 o 
 
 oo 
 
 ooo 
 
 00 00 
 
 o 
 
 
 
 
 1 1 
 
 4- 1 
 
 ."Ti 
 
 1 1 
 
 
 II 14- 
 
 + 
 
 
 
 
 
 
 
 
 
 tK-f-aJi- f 
 
 
 
 
 
 
 
 
 o 
 
 CO i-H 
 
 rH rH 
 CM rH rH 
 
 *?X> * rH tO 
 
 
 
 . 
 
 
 
 
 
 80 
 
 888 
 
 888 88 
 
 
 
 s 
 
 
 !*.'()! 
 
 nj. 
 
 ;ww 
 
 o o 
 
 ooo 
 
 o' o' o' o" o 
 
 
 
 
 
 
 
 
 ++ 
 
 1 1 1 
 
 * 
 
 
 
 
 
 
 
 
 
 
 
 
 
 X 
 
 
 
 
 
 
 
 
 
 
 o 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 N 
 
 
 
 x, t*) ^^ a 
 
 
 
 
 
 
 
 1 
 
 
 
 II 1 1 '3 
 
 
 
 
 CM * 
 
 CO 
 
 <N 
 
 CO 
 
 IM CO 
 
 COlO 
 
 CM Tt* CO CM IO CM -^ cj^CM 
 
 5 
 
 
 
 
 
 
 
 
 
 -) (- -)-+ ++4- H h+ 1 + 4- 
 
 4- 
 
 
 
 IN <M Tf 
 
 TT 
 
 3 
 
 5 
 
 COCO 
 CM CO 
 
 coco cc. 
 
 (M IN CO 
 
 CMC-1CO CMCO CMCMCO (NCMCO -<N CM 
 
 co 
 <r 
 
 
 
 V 
 
 (=- 
 
 H 
 
 
 
 
 V V 
 
 
 
 
 B- "B- 
 
 p- 
 
 
 - 
 
 R ' 
 
 * 
 
 S- K- K- 
 
 (S C 
 
 s* 
 
No. 3.] 
 
 MINOR PLANETS LEUSCHNER, GLANCY, LEVY. 
 
 117 
 
 
 
 
 
 > 
 
 d c^ o O t* t* 
 
 X 
 
 ic r^ oi .-i r-4 o 
 
 11+ + 1 4- 
 
 t 
 
 t-- to c^ o co 
 
 C5 cc ^ 00 t i ^ 
 
 CC O O CO CD C** 
 
 X 
 
 O 00 * O O O 
 
 . 
 
 l-f 1 1 + 1 
 
 i 
 
 e o to S t-~ oc 
 
 1 
 
 C " < O O O O 
 
 11+ + 1 + 
 
 t" 
 
 e o 8 8 8 o 
 
 Ir 
 
 o o o o o 
 
 1 + 1 1 + 1 
 
 ' 6 
 
 CO CO 
 OC CC t~ CC Tf 
 
 
 c'o" o o' o' o' 
 11+ + 1 + 
 
 C 
 
 
 
 
 S 
 
 
 
 s 
 
 
 + + 7 
 
 
 ^- i- *^s 
 
 
 v v v ? 
 
 B- P* F *=- 
 
 
 O (T- C- 
 
 
 .-.I 
 
118 
 
 MEMOIRS NATIONAL ACADEMY OF SCIENCES. 
 
 TABLE XXIXc. 
 
 [Vol. XIV. 
 
 Unlt-1" 
 
 
 
 w-i 
 
 ;-* 
 
 
 Cos 
 
 
 
 
 
 U)0 
 
 
 
 
 
 W 
 
 w<> 
 
 w 
 
 w 
 
 
 +20+24 
 
 
 - 90.5 
 
 + 302. 7 
 
 - 478. 2 
 
 
 
 
 
 2t+40+44 
 
 
 - 26.6 
 
 + 125. 5 
 
 - 270. 3 
 
 
 
 
 5 
 
 20+24 
 
 + 589.8 
 
 - 1571.9 
 
 + 1680 
 
 
 
 - 1.82 
 
 +12.00 
 
 
 1 
 
 
 
 
 
 
 + 0.42 
 
 - 2.10 
 
 
 +40+44 
 
 
 + 616 
 
 - 3638 
 
 +11175 
 
 
 - 0.42 
 
 + 2.10 
 
 
 2+20+24 
 
 
 + 23 
 
 - 161 
 
 + 439 
 
 
 
 
 
 2 +69+64 
 
 
 + 219 
 
 - 1451 
 
 + 4616 
 
 
 
 
 *' 
 
 219+ 4 
 
 - 106.1 
 
 + 360 
 
 - 517 
 
 
 
 + 1.81 
 
 -10.64 
 
 
 t+ * 
 
 
 - 43 
 
 + 161 
 
 - 269 
 
 
 - 0.08 
 
 + 0.45 
 
 
 s+49+34 
 
 
 - 760 
 
 + 3906 
 
 -10778 
 
 
 + 0.08 
 
 0.45 
 
 
 2f+29+34 
 
 
 + 52 
 
 - 143 
 
 + 87 
 
 
 
 
 
 2 t +69+54 
 
 
 - 314 
 
 + 1874 
 
 - 5403 
 
 
 
 
 f 
 
 
 
 
 
 
 -0. 317 
 
 + 1.63 
 
 
 
 49+44 
 
 -1679 
 
 + 7272 
 
 -13527 
 
 
 -0. 633 
 
 +10. 02 
 
 
 
 s+20+24 
 
 
 + 274 
 
 - 63 
 
 
 
 - 1.20 
 
 
 
 +66+64 
 
 
 - 3474 
 
 +29267 
 
 
 
 + 3.60 
 
 
 
 - +29+24 
 
 
 + 1156 
 
 - 2171 
 
 
 
 - 2.40 
 
 
 
 2t 
 
 
 
 
 
 
 - 0.21 
 
 
 
 2f +49+44 
 
 
 + 180 
 
 + 113 
 
 
 
 + 0.21 
 
 
 
 2e+89+84 
 
 
 - 1375 
 
 +11897 
 
 
 
 
 
 if 
 
 4 
 
 
 
 
 
 +0. 227 
 
 - 1.30 
 
 
 
 40+34 
 
 +3690 
 
 -12966 
 
 +19401 
 
 
 +0. 340 
 
 -19. 92 
 
 "" 5 ~ 
 
 
 +20+ J 
 
 
 + 222 
 
 - 1234 
 
 
 
 + 1.96 
 
 
 
 +29+34 
 
 
 - 769 
 
 + 2197 
 
 
 
 + 0.01 
 
 % bS 
 
 
 e+69+54 
 
 
 + 9240 
 
 -70866 
 
 
 
 - 7.25 
 
 
 
 - +29+ 4 
 
 
 - 646 
 
 + 1806 
 
 
 
 + 5.27 
 
 
 
 2s+ J 
 
 
 + 99 
 
 - 444 
 
 
 
 + 0.04 
 
 
 
 2t+40+34 
 
 
 - 109 
 
 - 922 
 
 
 
 - 0.04 
 
 
 
 2f+40+54 
 
 
 - 846 
 
 + 4256 
 
 
 
 
 
 
 2J+80+74 
 
 
 + 4012 
 
 -31827 
 
 
 
 
 
 l" 
 
 
 
 
 
 
 -0. 039 
 
 + 0.24 
 
 
 
 40+24 
 
 -1780 
 
 + 4725 
 
 - 5354 
 
 
 -0. 039 
 
 +10.42 
 
 s '3 ~ 
 
 
 f+20+24 
 
 
 + 499 
 
 - 649 
 
 
 
 - 0.32 
 
 
 
 t+60+44 
 
 
 - 5930 
 
 +40905 
 
 
 
 + 2.86 
 
 
 
 -+20 
 
 
 
 
 
 
 -2.54 
 
 
 
 2+ 24 
 
 
 - 65 
 
 + 285 
 
 
 
 
 
 
 2 t +40+44 
 
 
 + 980 
 
 - 4150 
 
 
 
 
 
 
 2 t +89+64 
 
 
 - 2890 
 
 +20791 
 
 
 
 
 
 j 1 
 
 49+34 -2 
 
 - 101 
 
 + 493 
 
 - 1128 
 
 
 
 + 0.11 
 
 
 
 +29+24 
 
 
 + 587 
 
 - 2983 
 
 
 
 
 
 
 t+69+54-J 
 
 
 - 193 
 
 + 1759 
 
 
 
 + 0.14 
 
 
 
 - +20+ J-2 
 
 
 
 
 
 
 - 0.14 
 
 
 
 2+ 4+2" 
 
 
 - 192 
 
 + 705 
 
 
 
 
 
 
 2 t +40+44 
 
 
 + 298 
 
 - 1876 
 
 
 
 
 
 
 2+80+74-J 
 
 
 - 65 
 
 + 616 
 
 
 
 
 
 
 
 mf 
 
 m" 
 
No. 3.] 
 
 MINOR PLANETS LEUSCHNER, GLANCY. LEVY. 
 
 119 
 
 TABLE XXIXc Continued. 
 
 
 
 fnit-l" 
 
 
 Cos tc 
 
 w 
 
 - 
 
 
 $ + 0+ 4 
 
 - 31.4 
 
 + 131. 
 
 - 255 
 
 
 
 E+30+3J 
 
 - 48.0 
 
 + 193. 6 
 
 - 360 
 
 
 "" J 
 
 +50+54 
 
 - 15.2 
 
 + 81.7 
 
 - 201 
 
 
 
 [E+79+74 
 
 5.2 
 
 + 34.7 
 
 - 107 
 
 1J 
 
 $+30+34 
 
 + 1304 
 
 - 8173 
 
 +30282 
 
 
 -$+ 8+ 4 
 
 - 196 
 
 + 506 
 
 - 598 
 
 
 $+ 0+ 4 
 
 + 34 
 
 - 146 
 
 + 292 
 
 
 $+50+54 
 
 + 356 
 
 - 2212 
 
 + 6781 
 
 
 $+30+34 
 
 + 1 
 
 67 
 
 + 294 
 
 
 $e+70+7J 
 
 + 138 
 
 - 999 
 
 + 3437 
 
 ,/ 
 
 ( 
 
 ^+30+24 
 
 - 1361 
 
 + 7468 
 
 -25691 
 
 
 : 
 
 + 
 
 
 
 
 
 
 
 + 29 
 
 12 
 
 - 155 
 
 
 
 _|-50.|-4J 
 
 - 482 
 
 + 2635 
 
 - 7209 
 
 
 
 -i-39+44 
 
 + 52 
 
 - 197 
 
 + 280 
 
 
 $+70+64 
 
 - 207 
 
 + 1348 
 
 - 4176 
 
 j> 
 
 $t+ 0+ 4 
 
 - 625 
 
 + 3058 
 
 
 
 $+50+54 
 
 - 7151 
 
 + 70387 
 
 
 
 -$+30+34 
 $+30+34 
 
 + 5478 
 + 18 
 
 - 5874 
 + 1924 
 
 
 
 $+70+74 
 
 - 2111 
 
 + 17665 
 
 
 
 -$+ 0+ 4 
 
 + 187 
 
 - 590 
 
 
 f,e . 
 
 $+90+94 
 
 - 771 
 
 + 6931 
 
 
 / 
 
 
 + 0+24 
 
 - 231 
 
 - 1142 
 
 
 
 
 
 +17640 
 
 -159928 
 
 
 
 
 
 +30+24 
 
 - 9842 
 
 + 1346 
 
 
 
 i 
 
 +30+44 
 
 - 892 
 
 + 3699 
 
 
 
 
 e+30+24 
 
 + 106 
 
 - 2494 
 
 
 
 j 
 
 j-j-70+64 
 
 + 5918 
 
 - 45149 
 
 
 
 
 
 + 
 
 
 
 
 
 $+90+84 
 
 + 2513 
 
 - 20914 
 
 
 ^ 
 
 ^ 
 
 k+ 0+ 4 
 
 - 507 
 
 + 2729 
 
 
 
 1 
 
 +50+34 
 
 -10202 
 
 + 84314 
 
 
 
 1 
 
 +30+ 4 
 
 + 1055 
 
 - 678 
 
 
 
 j 
 
 - 0+ 4 
 
 - 100 
 
 + 387 
 
 
 
 j 
 
 +30+34 
 
 + 871 
 
 - 2817 
 
 
 
 
 
 +70+54 
 
 - 4065 
 
 + 27951 
 
 
 f 
 
 $+ 0+ 4 
 
 + 601 
 
 - 3122 
 
 
 
 i 
 
 +50+44 
 
 - 423 
 
 + 4435 
 
 
 
 
 
 L QA J^O J V 
 
 f |~*X/-^fc d ~~~ 
 
 + 285 
 
 - 356 
 
 
 
 j 
 
 +30+34 
 
 + 426 
 
 - 2410 
 
 
 
 . 
 
 +70+64-2" 
 
 - 108 
 
 + 988 
 
 
 
 ~ n 
 
 *+ -2 
 
 - 106 
 
 + 402 
 
 
 
 
 m' 
 
 
 
 tnlqii 
 
 fn; 
 
 
120 
 
 MEMOIRS NATIONAL ACADEMY OF SCIENCES. 
 
 [Vol. XIV. 
 
 TABLE XXIXc Continued. 
 
 i- 
 
 Unlt-l' 
 
 
 Cos 
 
 
 .. 
 
 
 
 w - 
 
 
 
 
 * 
 
 w 
 
 
 
 ^ 
 
 , 
 
 
 
 f 
 
 20+24 
 
 + 1568 
 
 - 8912 
 
 
 +3.6 
 
 -23.3 
 
 
 
 60+64 
 
 + 5879 
 
 - 31559 
 
 
 +3.6 
 
 -23.3 
 
 
 gY 
 
 20+ 4 
 
 - 2385 
 
 + 11662 
 
 
 -4.7 
 
 +26.6 
 
 
 
 20+34 
 
 + 2238 
 
 - 1015 
 
 
 -2.6 
 
 +18.4 
 
 
 
 60+54 
 
 -21644 
 
 +102003 
 
 
 -9.9 
 
 +59.1 
 
 
 1) I)' 3 
 
 20 
 
 
 
 
 -0.2 
 
 + 1.7 
 
 
 
 20+24 
 
 - 1723 
 
 - 7588 
 
 
 +4.0 
 
 -27.1 
 
 
 
 60+44 
 
 +25396 
 
 -103013 
 
 
 +7.6 
 
 -43.2 
 
 
 V s 
 
 20+ 4 
 
 - 1160 
 
 + 5960 
 
 
 -1.4 
 
 + 8.3 
 
 
 
 60+34 
 
 - 9257 
 
 + 31500 
 
 
 -1.4 
 
 + 8.3 
 
 
 } ijj 
 
 60+54 -S 
 
 + 1040 
 
 - 6697 
 
 
 +0.3 
 
 - 2.1 
 
 
 
 20+24 
 
 - 5354 
 
 + 25370 
 
 -61855 
 
 
 
 
 
 sew 
 
 
 
 
 
 
 
 j 1 tf 
 
 20+ 4 
 
 + 2492 
 
 - 12023 
 
 
 
 
 
 
 20+24 -2 
 
 - 53 
 
 + 989 
 
 
 -0.1 
 
 + 0.6 
 
 
 
 60+44 -J 
 
 - 1285 
 
 + 7413 
 
 
 -0.1 
 
 + 0.6 
 
 
 
 
 1 j j t> 
 
 
 L" *-h 
 
 
 
 
 
 (0-0 ) sin 
 
 tui 
 
 4 
 
 t -ft 
 
 
 
 
 
 ! Writ 
 
 
 
 
 
 
 
 i) 
 
 20+24 
 
 
 
 
 
 - 0.55 
 
 +3.40 
 
 
 swi 
 
 
 
 
 
 
 
 1 
 
 20+ 4 
 
 - i <!.'.': 
 
 
 
 
 + 0.41 
 
 -2.74 
 
 jf 
 
 40+44 
 
 4 (-.". 
 
 
 
 
 - 3.12 
 
 
 
 +20+24 
 
 
 
 
 
 - 1.10 
 
 
 
 -+20+24 
 
 
 
 
 
 - 1.10 
 
 
 Tj tf 
 
 4 
 
 - 569.95 
 
 + 2421. 1 
 
 - 4950 
 
 +0.45 
 
 + 5.94 
 
 
 
 40+34 
 
 
 
 
 
 - 5.75 
 
 
 
 +20+ 4 
 
 i ~ii~ 
 
 
 
 
 + 0.20 
 
 
 
 +20+34 
 
 '''*(' 
 
 
 
 
 + 0.82 
 
 
 
 -+20+ 4 
 
 
 
 t* ^- $ 
 
 +*| 
 
 + 1.01 
 
 
 ,/J 
 
 40+24 
 
 'i(l! 
 
 
 t +e 
 
 U-: -j ft 
 
 -i i 
 
 + 2.55 
 
 
 
 +20+24 
 
 soot 
 
 
 I.""' ^ ft 
 
 ' 1 -f 
 
 - 0.15 
 
 
 
 -+20 
 
 
 
 
 
 - 0.15 
 
 
 
 Si'U; 
 
 
 
 
 
 
 
 
 (0-0 ) a coa 
 
 i'.'it' 
 
 
 "i U~t * 
 
 U* 
 
 
 
 
 <*f.<; 
 
 i"^82 
 
 
 i. tt fti 
 
 1 4 - 
 
 
 
 >) "?' 
 
 J 
 
 di'f" 
 
 
 U' '. * 
 
 
 - 0.26 
 
 
 
 HK*' 
 
 801 
 
 
 '** - L3 --ft 
 
 
 
 
 '" 
 
 ':( 
 
 
 
 
 
 + 0.20 
 
 
 
 
 
 TO' 
 
 
 
 m' 2 
 
 
 COMPARISON OF TABLES. 
 
 As a computer would discover in constructing tables, and as will be evident from an appli- 
 cation of the method to a planet, the coefficients in Table II and others of the same form are 
 given with unnecessary accuracy. Although so many digits would never be required, except 
 in a much more exhaustive development, they are given, for completeness, as they resulted 
 from computation. 
 
 In all the tables whose constructions involve the multiplication of trigonometric series, the 
 errors are difficult or impossible to determine. Although v. Zeipel's manuscript, which the 
 author generously furnished for comparison, is of assistance, the computations are not entirely 
 parallel, and comparison is not always possible. Many of the computations are so long and 
 
NO. 3.] MINOR PLANETS LEUSCHNER, CLANCY, LEVY. 121 
 
 complicated that the origin of certain discrepancies is obscure. Aside from possible errors of 
 calculation, differences are due to the independent adoption of the highest powers of m', w, ij, 17', f, 
 and the number of arguments in a given series or product of series. In most cases our series 
 are more complete than v. Zeipel's. Whether or not the extension of the tables increases the 
 accuracy of the result remains to be seen from future applications of the theory. 
 
 Tables II-XV. -The discrepancies seem to be due to v. Zeipel's errors of calculation and to 
 their subsequent effects. The larger number of these errors have been traced in the manuscript. 
 
 Tables XVI, XVII check satisfactorily. 
 
 Table XVIII. The bracketed quantities in the first three columns are in error through 
 previous discrepancies. We did not discover the source of the general disagreement in terms 
 of the third degree, second order in the mass. These terms do not affect v. Zeipel's subsequent 
 tables, since they are of order higher than have been included. 
 
 Tables XIX, XX agree satisfactorily. 
 
 Table XXI. The discrepancies are numerous and their origin is obscure because of the 
 very long computation involved. In addition to performing a complete duplicate computation, 
 the terms of first degree and a part of the computation of second degree terms were checked 
 by the solution of the differential equation in the form given in Z 64. With the exception of 
 three or four single instances, the discrepancies occur in two groups, having the following 
 probable explanations. The neglect of the term 
 
 in Z 65, eq. (109), by v. Zeipel accounts for one group of differences. The other group can be 
 
 fairly well explained by an error in the addition of second order terms in +- fa to #, -^#.. 
 
 A & 
 
 Assuming that for these terms he added w<t>, and, correcting his table, three discrepancies are 
 removed and two others are improved. 
 
 Table XXII. Considering the disagreements in Table XXI, Table XXII checks satis- 
 factorily. 
 
 Table XXIII-XXVn. These tables, like II-XV, are simple in construction, and the 
 discrepancies are due to errors of calculation, or they are the result of previous ones, with the 
 exception that some quantities have different numerical values because they are more inclusive. 
 The latter have been indicated by ( ). 
 
 Table XXVLLi. The discrepancies arise from the quantities in parentheses in Table XXVEI. 
 The omission of the term depending upon the inclination is justifiable in view of its magnitude. 
 
 Table XXIX. The discrepancies are numerous and striking, but, since v. Zeipel does not give 
 the formulae of computation, they remain unexplained. The remark is made (Z 77), "Die 
 Berechnung der Funktion [(1 e cos ) ( W 3 W 3 ")], welche eine sehr komplicirte war, wird hier 
 nicht im Einzelnen mitgetheilt." For this reason the development of the formulae which we 
 used has been included and the auxiliary functions 2[TJ, W 3 , [(le cos e) W s "] have been 
 tabulated. The differences are not serious because of the high rank of the function. Our 
 table is deficient in certain terms whose computation would be long and the omission of which 
 is justifiable in view of their magnitude. 
 
 ',i-ffttl -jilj i: i :;// i <: r\ v;?"t nvtA'fi 
 
 PERTURBATIONS OF THE MEAN ANOMALY. 
 
 For clearness some of v. Zeipel's developments will be amplified and repeated in an order 
 which we found more convenient. 
 
 The determination of the disturbed mean anomaly is accomplished with the integration 
 of Z 9, eq. (47), (which implicitly contains Z 8, eq. (38)). By Z 9, eq. (43), 
 
 d = %(e-esms)-g' = 1 s g-g' 
 
 The differential equation is repeated in Z 78, eq. (124), in which is emphasized the fact that 
 
 d W 
 the arguments are functions of both e and 0, as is the case for r- 
 
122 MEMOIRS NATIONAL ACADEMY OF SCIENCES. [voi.xiv. 
 
 If we observe the character of as it is expressed in the definition and recall that we have 
 admitted trigonometric terms in 0, multiplied by t, it is evident that this argument, which is 
 a function of the disturbed positions of the planet and Jupiter, is not periodic, but varies con- 
 tinuously with the time. In the foregoing equation g and g' can not be regarded as angles 
 which are always less than 360. contains, therefore, a nontrigonometric secular part in e 
 and a periodic part in 6 and s. 
 
 If we write 
 
 6=(0-[0]) + [S] 
 
 [6] contains the secular term in s as well as periodic terms. The segregation of terms of 
 different type can be made explicit by the introduction of 
 
 fe Z 78 ' ! 
 
 where i? is a function of s and d lt 2 , 6 3 are the periodic parts of ff [0], i. e., they are 
 entirely trigonometric functions of e. This covers the condition that 9 t can not include trigo- 
 nometric secular terms in e. By definition of tf and Q i 
 
 i 
 
 *? = [fi = [/??,)] - ^^ 
 
 ds [_dJ ds 
 
 where [n'tis*] is the long period term between Jupiter and Saturn. 
 
 The derivative of (125) is 
 KJ n,. <-u<>;-4 ; .fe'nnoionib ro quuig aiio aininmA laquhx .7 vu ,(60Ij ,p<j ,t .. :v IK 
 
 , 
 
 ^ = ^ + ^^ 
 or ds odds 
 
 <pT' ; ibe, be, bo, \ / be, be be, \d& 
 
 = I -^ * -J 2 4- Tr-5 4- 14-ll-J ' -4 A 4- \-r 
 
 \bs be bs / V d$ 5$ d?? /ds 
 
 Expanding F(6,s), eq. (124) in a Taylor's series in ascending powers of 0<and making the above 
 
 substitution for-y- (124) becomes (126), in which 
 hi) liiiv/ ,-ino v:j.vv--nq lo J(;j ';'". '>il) -JTB T->I!> in .noifjiliJ >?> lo snon-j at emh <nu 8?j! r >aBtjOTO8tL 
 
 j Q 
 
 From the Taylor's series T- is written m (127). This is the differential equation for tf, the 
 
 a 
 
 right-hand side of which can be computed. 
 
 Substituting 3- in (126) and equating functions of equal rank, we have the differential 
 
 ' TOD gJttlHil 
 
 equations (128j 128 3 ) for 0<, which can be integrated in succession. 
 
 Before integration we convert eqs. (128) into differential equations for ndz as follows: 
 
 ii'' r*Siou ifiiii ori.) lo i , .. , . ,. . . , Joo WIR 
 
 n8z = (7^3 - [n<52]) + [72^2] 
 
 ' r . , 00 /, , .N 
 
 = 7n?2j+ r^2 2 + n^2 3 H ------ \-[n8z] Z 88, eq. (144), 
 
 where n8z t is not only a function of first and higher orders in m', in which the lowest rank is i, 
 but is entirely trigonometric or periodic. Then 
 
 r iri* : 'f" o 
 
 2 r 
 Z 9, eq.(46) gives flfe-[nte]-r |^ x (*>) + 0, (&,e) +6 3 (&,e) + ...... +wi? sin e+ (n'8z' -[n'dz']) 
 
 and 
 
 
 [n52] = r f^{t>-| + [7i'fe']4-c'-//c] Z 88, eq. (145), 
 
 where it is to be noticed that [ndz], unlike [ W], is not free from terms in e. Subdividing the first 
 of these two equations according to rank, we have Z 79, eqs. (130), in which n'dz' + [n'dz'] can 
 be neglected. 
 
NO. 3.] MINOR PLANETS LEUSCHNER, GLANCY, LEVY. 123 
 
 Differentiating eqs. (130) partially with respect to e, substituting in eqs. (128), evaluating 
 the right-hand sides of eqs. (128), we have eqs. (131, 131,), in which the superscript indicates 
 that only terms of first order in the mass are included, and where the argument tf replaces 
 the argument 8. 
 
 For purposes of calculation, the integrations are arranged as follows: 
 
 In 
 
 + W 3 "+ F 4 ") 
 
 consider first only W t "+ W 3 " + W t " in the integration of eqs. (131). The integrations will 
 concern only part of the terms of first order in ndz l + nJiz 2 +ndz t . It is shown by v. Zeipel that 
 the integration for all three ranks can be performed conveniently at the same tune. Let this 
 part of the function be indicated by enclosing it in ( ). The integral 
 
 + 
 
 which is a trigonometric series, is given by Z 80, eq. (135), in which the coefficients L p . q are 
 defined by (136) and are easily derived from Table XXVII. The coefficients L p ^ are tabulated 
 in Table XXX. 
 
 The remaining terms of rank one which are of first order only, namely, ndz^ (ndz^), are 
 given by the first of Z 81, eqs. (137), in which TT,, IF,, [FJ, can be written by inspection 
 from Tables XVH, XVm, XIX, XXIIa, The function is tabulated hi Table XXXI. 
 
 The remaining terms of first order in ndz 2 and ndz 3 are given by the sum of Z 82, eqs. (139) 
 and (140). The function 
 
 ___ 
 
 is given in Table XXXH. 
 
 These developments complete ndz (1) within the limits of the tables, and we next consider 
 ndz (2) . We shall limit ourselves to functions in which the lowest rank is the first or second. 
 
 Consequently, ndz^ contributes nothing. 
 
 m 't 
 
 Anv function of second order in the mass and first rank must contain the factor r and in 
 
 itr 
 
 the given F (t>, e) this factor occurs only in Wf. We have, therefore, by Z 80, eq. (131,), for 
 one part of ndz^\ indicated by parentheses, 
 
 >) = f{(l -e cose) F^-tU -ecoas) W]}dt 
 This function is tabulated in Table XXXIII. 
 
124 
 
 MEMOIRS NATIONAL ACADEMY OF SCIENCES. 
 
 [Vol. XIV.. 
 
 
 co- 
 co 
 
 i* -T 
 
 * .0 
 
 CO 
 
 2 
 
 o 
 
 o co" 
 
 rH 
 
 CN 2 
 
 O 1 CO CO CS CM lO CO CO O CM O **< lO CO lOt* <M OS 
 CM CO CS CM CD wS ^* rH COCNlO 
 
 
 
 
 
 
 
 + 
 
 + 1 
 
 1 + 
 
 1 + 1 ++ 1 + 1 + ++ 11 1 1 + + 1 
 
 
 g 
 
 CO CO 
 
 OS OS 
 
 t^ 
 
 
 o 
 
 ' CO 
 
 CO CO 
 
 O CO r- 1 COOOCS CSrHOO ^eNt^-O CD t*rH iCW ^COCO COCOlCt^- 
 (>JOO COCNJlO lOO O3 rH iTiCOCO QOrHCOlOrHt--. 
 
 O> 
 
 
 
 
 OS rH^ i-HCO CO M C^ICS lOCOCO 
 
 
 
 
 
 rH N rH b 
 
 
 + 
 
 + 1 
 
 1 + 
 
 1+ I++I +1+ ++I.I 1 1+ +i 1+7 +11 + 
 
 
 i 
 
 CO lO 
 
 1O CO 
 
 CO 
 
 
 Ij 
 
 O CO 
 
 CO i-i 
 
 OOJO -^C^^iO <MCOCN C'-COC^ICTS CN OOrH HH > 1 C^^d O CO CO i 1 
 
 GO 
 
 
 CO 
 
 
 COf-H COCOC^ rHCOTt< lOfHrH rH O IOO3 C^rH ^COlCCO 
 ^ CNO MrH t^ lO COCO *OCOlO 
 Ci CO rH CO N t* 
 CO CO 
 
 
 + 
 
 + 1 
 
 1 + 
 
 1+ I++I +I+++II 1 1++I l + l +11 + 
 
 
 
 
 CO OS 
 
 rH CM 
 
 CD 
 
 
 CM 
 
 O OS 
 
 rH lO 
 
 OOO"^ OIOOO3 COOSCO CNOrHO *C t^-00 COt^- O CO^ CO W C^J t- 
 
 
 
 
 r-l 
 
 rHO *^COC^ iiO-CO rHlJSc^^T 1 C^ lO COM CM rH COOOrH 
 
 
 
 r-l 
 
 rH 
 
 iO -^CO COCO rH lO O rH t^- t^Ot 
 
 
 
 
 
 CO I s - C^ itrH^COMCO 
 
 ^* cs 
 
 
 + 
 
 + 1 
 
 1 + 
 
 i + i ++ i + i + ++ i i i i + + i +7+ii + 
 
 
 rH 
 00 
 
 CO C<> 
 
 OS CO 
 
 CO 
 
 to 
 
 CO 
 
 O CO 
 OS 
 
 CO CO 
 
 ^s 
 
 II1TJU5 rHt^-OiO fH-^CO O rH O rH -^ ^i CM COCO iQOOO CO'JJ-^cO 
 tN.^'cOlO^CO MrHCM OSrH frHCO 
 
 
 
 
 
 M CD ^ rH 
 
 
 
 
 
 ^ 
 
 
 i 
 
 1 + 
 
 1 + 
 
 1 + 1 ++II +1+++II 1 +++I +1++1+ + 
 
 
 rH 
 
 U5 t- 
 
 10 10 
 
 00 
 
 "5 
 
 rH 
 
 rH 
 
 CO O 
 
 OJCOO CO^rHCN I>- CO 1C COCNrH-^ O* COM COCO COCO^ rHCOrHO 
 
 CD US -^rHO OCS lO CMrH t-rHO CO 1 ^ COOO4 
 OOrHOarHOO COCOCOCNIOCMCD 
 
 
 
 
 
 1O rH rH rH C 
 lO C^J 
 
 
 + 
 
 1 + 
 
 1 + 
 
 1 + 1 ++ 1 + + 1 + + 1 1 i 1 ++ +1 + ! + t 1 + 1 
 
 
 CO 
 
 CO GO 
 
 TJ* CO 
 
 CO 
 
 
 rH 
 
 CO iO 
 CM CO 
 
 s s 
 
 CNCM rHiacp 1 -* 1 COOJCO lOCMt^-OS Oi 2^ U^rH CSCO I>-COOCN 
 
 f 
 
 
 rH 
 rH 
 
 15 
 
 t^- rH C^ s " CM rH CO 
 rH ^< 
 
 
 + 
 
 1 + 
 
 1 + 
 
 i i ++I++M+III i+i+i i+ i ^+7 
 
 
 S3 
 
 iO CO 
 
 *< CO 
 
 in 
 
 
 8 
 
 CO CO 
 r-i CO 
 
 CO CS 
 00 00 
 OS 
 
 COrHCM t^t-' COrH* rH OlO CO r-ICO t^CO COOSCC IO(MCO~ 
 1OOS (N*t*CO 1OCO1O CN lOC^l CO WCM rHO COOSCM COrHCM 
 
 
 
 rH 
 
 rH 
 
 rH 
 
 T)CO i-irH COCO rHrHO * CO-9- COt~ 
 IM t- rH rH CO *-' 
 rH 
 
 
 + 
 
 1 + 
 
 1 1 
 
 + 11 +1+ +11 + 1+ 1 +1 ++ 1 + 1 ++ 1 
 
 
 i 
 
 w 
 
 m cxi 
 
 rH __ _ 
 
 
 
 
 S 
 
 r-i r-i 
 
 SrHl-*- CO CO t 1 -^ COCO COrHrHiO t^ CO t CO COCOi 1 CMCMCDO 
 
 
 
 
 CO 
 
 
 
 
 
 rH 
 
 rH OS CO rH O 
 
 
 i 
 
 + 
 
 1 1 
 
 + 1 + 1 + + + ++ ++ 1 + + 1 l+l +J^ +11 + 
 
 
 1 
 
 tO 00 
 
 CO 00 
 
 ss 
 
 lO 
 
 00 
 rH 
 
 OOOlCJ CO OSt^- OSOCO CMCOrH CO ^O CO CMOSCO CM OrH 
 lOCOrH CO lOOS CMCOrH rHCOO rH COCO CO t-COlO b* rHCM 
 CM i ( ^ CO lOt^- rHCTS-^ lO^TCM ^ O rH t* TfCOrH O rH^ 
 
 
 
 f ( 
 
 
 O CMrHrHrHCO rH CMrHCO rHCOrHfiM^CO 
 
 
 
 
 
 rH rH kA rH 
 
 
 i 
 
 + 1 
 
 1 
 
 +11 1 +1 + 1 + +++ + ++ + 1 ++ + ^+ 
 
 
 
 id id 
 
 C~J CXI 
 
 CO CO 
 
 cIS 
 
 CO CO rHCOOOrH CC CO r^t^OO JA^ OO CXI (M OSCNfMCS 
 
 o 
 
 
 
 o o 
 
 
 
 
 
 rH rH 
 
 l>t~ COCO COCO <Nr-t-<M 
 
 
 
 
 
 CO CO 
 
 
 
 + 1 
 
 1 + 
 
 +111 ++ 1 ++I+I I++I 1 + ++ 1 1 
 
 
 
 
 
 O C 
 
 
 
 
 
 rH ft t t rH rHrHrHrH rHrHrHrH 
 ^ ^ ++ II ^^+11+ ^^ ^.^ ++ 1 1 
 
 
 
 e e 
 
 r-l rH 
 
 g g gggg(M (M gggg ggrHrH g g gggg 
 
 * 
 
 _^_^ 
 
 1 1 
 
 + 1 
 
 
 
 se 
 
 
 
 
 ^-.'ff-j* .... ggg .... g .. gg -g- .... 
 
 
 1 
 
 + 7 
 
 I 1 
 
 + '. i +1 + 1 L '. '. +1 + 1 '. +7 L '. + L i +1 + 1 
 
 
 s 
 
 s s 
 
 ~ 
 
 gj?S ggg JSSSS ?S SS SS SSS SSSff 
 
 
 o 
 
 *~o *~o 
 
 *~~^ *~~-i 
 
 ^"o *^o ^"o ^- s ^- ^^ **~Z- ^"tN s *w > *^ ^~o ^^o ^^e **"o ^~o ^~c ^*o *~~^- ^~^- ^~o ^o ^~o s ~i x ~-C ^^ **~ 
 
 
 l-f 
 
 MM 
 
 iJW 
 
 -~~ ^~ ^-~ - ~- -- ^~, jt-3 -^ j^ !*-C i-C ;-*; Ik^ <; -C -^ -C ^ ^ -; -c ^ *-:;--- 
 
 
 
 
 
 (fl JO^DB^ 
 
 
 H 
 
Vo. 3.] 
 
 MINOR PLANETS LEUSCHNER, GLANCY, LEVY. 
 
 125 
 
 
 
 
 
 17.3TX ajaT 
 
 
 
 00 
 
 rH 
 
 
 {'''/***;- 
 
 i-. eo 3? 
 
 I-*- CO CO 
 
 CM t * O 
 
 rH 
 
 m 
 
 OJ CO iS Ol 
 
 rH rH, 
 
 -. 
 
 1 + 1 
 
 1 1 + + 
 
 + 
 
 41 1 + 
 
 "" iV* 4 tV--*!; 
 
 1^5 i** i& 
 
 O 01 00 
 
 S3S2 
 
 
 
 _- c^> oo 
 
 - .\ll 
 
 
 t i ( N 
 
 
 CO 
 
 . 
 
 1 47 
 
 1 1 44- 
 
 + 
 
 + 1 1 + 
 
 
 sss 
 
 rH rH rH 
 
 Bo^0 
 
 I 
 
 &* ro O5 iO 
 " CO d rH 
 
 
 CO t- 
 
 CM 
 
 
 00 O 
 
 _ 
 
 H 
 
 
 
 
 -. r 
 
 i + i 
 
 1 1 44 
 
 + 
 
 II 1 + 
 
 
 -..- 
 
 " 
 
 
 OS t- 
 
 *? fl "*" 
 
 rH t2 2 
 
 ^H i ^ s 
 
 
 r^ C 1 ! CO 'i' 
 
 
 ^ 
 
 ^ 
 
 
 S S 
 
 
 1 + 1 
 
 i i ++ 
 
 4 
 
 II 1 + 
 
 t -4- ? 
 
 
 
 
 
 WU*I -i- V-i-j 
 
 00 iC 
 CC Cl 
 
 OO * CO T 
 
 r t^- ifi- 10 
 
 
 
 sr 3 o 
 CO O) ' 
 
 
 
 
 >a 
 
 "& *& 
 
 
 OS 
 
 
 
 eo co 
 
 
 
 
 
 
 - *_ , - Hi* 
 
 1 + 1 
 
 1 44 + 
 
 + 
 
 1 J- ' + 
 
 1 
 
 Sig 
 
 seS 
 
 1 
 
 CM t- 2 s5 
 
 ^H) 
 
 D 
 
 r^ 
 
 *|So5 
 
 H 
 
 " S* 00 
 
 JS 
 
 rH 
 
 
 
 rH 
 
 a 
 
 
 
 
 
 08 
 
 1 44 
 
 1 + + + 
 
 4 
 
 1 + $ 
 
 
 
 
 
 
 ^> 
 
 
 
 
 
 .3 r.000fi-- 
 
 t-t CS -n* 
 O CM CO 
 
 CC 5S r^ 
 
 tO t^ t~~ 
 t- OS rH 
 
 CO lO 
 
 1 
 
 XX OO rH 
 
 C CM CO 
 r- CO -V 
 cp C5 
 
 55 O 
 
 
 
 s 
 
 
 
 " 
 
 1 
 
 1 + + 
 
 1 + 1 
 
 4 
 
 1 4 II 
 
 | 
 
 
 
 
 
 .3 
 
 SI 
 
 O CC 
 
 CO OO CO O 
 t^. t*- ^^ Ou 
 M O 5^ -H 
 
 rH kO CC 
 rH i-^ CO 
 
 1 
 
 P CM * 
 
 C7S CO t~ 
 
 **** CM O 
 
 5S 
 
 1 
 
 
 
 
 
 9 
 
 1 1 
 
 1 1 + 1 
 
 1 
 
 + + 1 
 
 fjsti; Z tv-t-f-' 
 
 
 
 
 
 
 
 ?,2g 
 
 <N -H OO 
 
 v * <y 
 
 * 
 
 "" Z 3 
 
 9 
 
 CC 1O O 
 
 co r- CM 
 
 
 >o 
 
 ~. ~ ^ 
 
 "3 
 
 a 
 
 
 
 
 
 1 4 1 
 
 1 1 1 
 
 1 
 
 1 1 1 
 
 
 
 
 
 
 
 B 
 
 
 
 
 
 a ;.,; 
 
 
 
 
 
 
 S 
 
 rH rH 
 
 liii 
 
 
 ^* ^* rH rH 
 
 
 
 4 1 
 
 ~ x, 
 
 i + 1 + 
 
 
 + 1 1 + 
 
 i 
 
 
 
 
 
 C3 
 
 
 iZZ 
 
 
 
 S. 
 
 
 4-774 
 
 
 
 .2 
 
 CM* sr 
 
 + -, 1 
 
 g s s e 
 1 1 1 1 
 
 
 "i"? r7r7 
 II +1 
 
 3 
 a 
 
 s s e 
 
 
 1? 
 
 8 8 
 
 S 
 
 I I 1 
 
 s e e" 
 
 rH T-^ ^^ ^H 
 
 4141 
 s e s s 
 
 1 
 
 8 
 
 rH rH . . 
 
 +1 '. '. 
 
 s e s e 
 
 r 
 1 
 
 i o o 
 
 I I S I 
 
 | 
 
 CO 
 
 CI 
 
 ^ ^ -^ 
 
 ** ** ^ ^ 
 
 '^ 
 
 i*-i ."*^ N 1 ? [^ 
 
 
 
 
 
 
 
 ^= 
 
 
 
 
 
 c_, 
 
 
 
 
 r" JaptU 
 
 
126 
 
 MEMOIRS NATIONAL ACADEMY OF SCIENCES. 
 TABLE XXXI. 
 
 [Vol. XIV 
 
 Unit-l" 
 
 
 Sin 
 
 *-. 
 
 
 
 
 . 
 
 w 
 
 w* 
 
 
 
 +2,5+24 
 
 + 294.89 
 
 740.6 
 
 + 734 
 
 
 B 
 
 
 
 
 
 
 
 
 +4,5+44 
 
 - 839. 5 
 
 + 3495 
 
 - 6224 
 
 
 
 2+2<5+24 
 
 - 147. 4 
 
 + 517 
 
 - 737 
 
 
 ft 
 
 e +4 
 
 
 
 
 
 
 +4,5+34 
 
 + 1229.8 
 
 - 4069 
 
 + 5671 
 
 
 j) 3 
 
 - +20+24 
 
 + 784 
 
 (- 3570) 
 
 (+ 10522) 
 
 
 
 +2,5+24 
 +6,5+64 
 2t+40+44 
 
 - 202 
 + 2940 
 + 415 
 
 (- 1657) 
 (- 17009) 
 (- 2587) 
 
 (+ 13183) 
 [(+ 43527)] 
 (+ 6440) 
 
 
 
 2f 
 
 
 - 192 
 
 + 705 
 
 
 i) i)' 
 
 - +2*+ 4 
 
 - 2386 
 
 (+ 11567) 
 
 (+ 37527) 
 
 
 
 +2,5+34 
 
 + 1492 
 
 (- 968) 
 
 (- 12562) 
 
 
 
 +20+ 4 
 +6,5+54 
 
 - 1962 
 - 8658 
 
 (+ 9257) 
 (+ 42767) 
 
 (- 23263) 
 (- 92732) 
 
 
 
 2+40+34 
 
 - 615 
 
 (+ 3264) 
 
 (- 6905) 
 
 
 
 2 +4 
 
 
 + 142 
 
 r- 605 
 
 
 B 
 
 - +2,5 
 
 + 1634 
 
 - 7081 
 
 + 16199 
 
 
 
 +20+24 
 
 - 861 
 
 - 3794 
 
 + 22127 
 
 
 
 +60+44 
 
 + 6349 
 
 - 25753 
 
 + 45318 
 
 
 f 
 
 - +20+ 4-2 
 
 + 866 
 
 - 4260 
 
 + 10988 
 
 
 
 +6,5+54-2 
 
 + 260 
 
 - 1674 
 
 + 5101 
 
 
 
 +20+24 
 
 - 2677 
 
 + 12681 
 
 - 30930 
 
 
 B 
 
 +40+44 
 
 + 5907 
 
 - 11149 
 
 
 
 
 - +4,5+44 
 
 - 269 
 
 + 5158 
 
 
 
 
 +8<5+8j 
 
 -11300 
 
 + 76249 
 
 i 5 S 
 
 
 B B 
 
 +40+54 
 
 -11449 
 
 + 42212 
 
 
 
 
 +40+34 
 
 -11270 
 
 + 951 
 
 
 
 
 _ +4,5+34 
 
 + 1744 
 
 - 23941 
 
 
 
 
 +80+74 
 
 +50005 
 
 -304611 
 
 
 
 ij ij' 2 
 
 +40+44 
 
 +26091 
 
 - 71730 
 
 
 
 
 +4,j+24 
 
 + 3985 
 
 + 16118 
 
 
 
 
 _ +4,5+24 
 
 - 3137 
 
 + 35021 
 
 
 
 
 +80+64 
 
 -73583 
 
 +400009 
 
 
 
 ij" 
 
 4-4^^-3J 
 
 -13756 
 
 + 22165 
 
 
 
 
 - +4^+ 4 
 
 + 3317 
 
 - 18452 
 
 
 
 
 +8,5+54 
 
 +36006 
 
 -172164 
 
 
 
 ?i) 
 
 +4^+34-2 
 
 - 1707 
 
 + 13125 
 
 
 
 
 +4^+342 
 
 - 2112 
 
 + 15096 
 
 
 
 
 +8,5 +74 2 
 
 - 2381 
 
 + 18919 
 
 
 
 
 +4^+44 
 
 +14204 
 
 - 88026 
 
 
 
 j 1 T)' 
 
 +4,5+44-2 
 
 - 554 
 
 + 140 
 
 
 
 
 +4^+242 
 
 + 3545 
 
 - 22885 
 
 
 
 
 +80+64-2 
 
 + 3827 
 
 - 27870 
 
 
 
 
 +40+34 
 
 -17503 
 
 + 99584 
 
 
 
 
 +(<5-i5 ) C06 
 
 
 
 
 
 1? 
 
 
 
 - 767. 7 
 
 + 2821 
 
 5210 
 
 
 1}' 
 
 + 4 
 
 + 570. 
 
 2421 
 
 + 4950 
 
 
 1* 
 
 2< 
 
 - 384 
 
 + 1410 
 
 - 2605 
 
 
 n i 
 
 2+ 4 
 
 + 285 
 
 - 1211 
 
 + 2475 
 
 
 rf 
 
 
 
 - 6624 
 
 + 47448 
 
 
 
 Y 
 
 + 4 
 
 [+17970] 
 
 [-120603] 
 
 
 
 
 + 4 
 
 + 8984 
 
 - 60301 
 
 
 
 i?" 
 
 +24 
 
 -10478 
 
 + 70250 
 
 
 
 
 f 
 
 -25564 
 
 +157424 
 
 
 
 >/* 
 
 + 4 
 
 +15678 
 
 - 94846 
 
 
 
 ,7*5 
 
 + 4 +1 
 
 -22012 
 
 +121258 
 
 -359162 
 
 
 
 e 
 
 +25565 
 
 -157424 
 
 [+511232] 
 
 
 /" ^' 
 
 +.T 
 
 +12048 
 
 - 76364 
 
 +251640 
 
 
 
 + 4 
 
 -23524 
 
 +150306 
 
 -498328 
 
 
 
 
 m' 
 
 
NO. s.] MINOR PLANETS LEUSCHNER, GLANCY, LEVY. 127 
 
 TABLE XXXII. 
 
 
 >o->'.-rt-{<' e; r*^ 
 
 Unlt-l". 
 
 
 
 
 r 
 
 
 oox 
 
 f'r% 
 
 !f 
 
 
 
 Sin 
 
 it* 
 
 . 
 
 
 
 _j-2i>+2J 
 
 ' - ' 294. 9 
 
 + 1036 
 
 " *" * 
 
 , 
 
 
 
 + 384 
 
 - 1410 
 
 *.j-wiT 
 
 nAfusi }.ij ( 
 
 2+2tf+2J ' ''' 
 
 + 1679 
 74 
 
 - 10348 
 
 + 149 
 
 '! ma 
 
 isftil'i fiiijJu 
 
 T .vti'io i^iTt i-i 
 
 
 
 
 /"" 
 
 t + ^ 
 
 - 285 
 
 + 1211 
 
 
 
 
 - 2460 
 
 + 13067 
 
 
 ((!> "' t\'j HfX 
 
 -' ' ' ,^ 
 
 
 
 
 ? 
 
 +2i>+2J 
 
 - 101 
 
 - 883 
 
 
 
 +2<>+2J 
 
 - 978 
 
 + 6459 
 
 
 fife ' p'J KK) 
 
 +60+64 
 
 - 8820 
 + 424 
 
 (+ 77487) 
 - 1332 
 
 
 
 2f 
 
 [+ 96] 
 
 I- 352] 
 
 
 ff' 
 
 / (1* t ' 
 
 - 2068 
 
 + 8418 
 
 
 
 -i-2i>-|-3J 
 
 - 1492 
 
 + 2460 
 
 
 
 fe.Q_l_R 1 
 
 c ~r "" i J 
 
 + 2280 
 + 25974 
 
 12618 
 (-206223) 
 
 i to ic 
 
 r* baa ooil 
 
 9 jf 
 
 - 615 
 t- 7 1] 
 
 + 1420 
 [+ 303] 
 
 
 1* 
 
 - +2* 
 
 + 1634 
 
 - 5447 
 
 
 
 +2tf+2J 
 
 + 861 
 
 + 2933 
 
 
 <? ' jW frt> 
 
 j-6fl-|-4j 
 
 - 19047 
 
 [+134400] 
 
 
 / 
 
 +2J+ 4 Z 
 
 + 866 
 
 - 3394 
 
 
 (j . * . * 
 
 ^_6tf+5J S 
 
 - 780 
 
 + 7362 
 
 
 
 j-j-2<>+24 
 
 + 2677 
 
 - 15358 
 
 
 ^ 
 
 +4tf+4j 
 
 - 5098 
 
 
 
 
 +4i>+44 
 
 + 4499 
 
 
 
 ^il y ' ~ 
 
 +8+8J 
 
 + 45200 
 
 
 r , 
 
 ?Y 
 
 + 40 + 5j 
 
 + 22898 
 
 
 
 v/ 
 
 -j-4t)^-3J 
 
 + 5322 
 
 
 
 '' ; '1 I ' ,=. 
 
 e+4t>+3J 
 
 - 11270 
 
 
 
 
 f -j-8iJ+7J 
 
 -200020 
 
 
 
 
 
 
 
 lr."W. 
 
 1 f 7 * 
 
 +4t5+4J 
 
 - 52182 
 
 
 
 
 +4i?+2^ 
 
 + 2712 
 
 
 ^jiixiu 
 
 
 +4^+2^ 
 
 + 4408 
 
 
 
 
 +8i9+64 
 
 +294332 
 
 
 
 V 
 
 +4i>+3J 
 
 + 27512 
 
 I Y. 7 7. .7.7 
 
 7. wI<! 
 
 
 _ _i-4^-i_ J 
 
 + 6634 
 
 
 
 
 +W+54 
 
 -144024 
 
 
 
 w 
 
 +4t>+3^-2 
 
 + 4022 
 
 
 r . BaM 
 
 
 +4J+34 2 
 
 - 3616 
 
 
 .<! ni f-. 
 
 
 +8t?+7*l 2 
 
 + 9524 
 
 
 
 
 +4d+4^ 
 
 - 28408 
 
 
 
 . r 
 
 
 
 
 
 
 y* f 7 
 
 e +4$+4j 'Z 
 
 + 1108 
 
 
 
 
 +4i>+2J 2 
 
 + 7090 
 
 
 
 
 +8i>+6J 2 
 
 - 15308 
 
 
 
 
 +40+34 
 
 + 35006 
 
 
 
 
 m' 
 
 
 ori.i 
 
 run- 
 
 -HT 
 
 
 
 The coefficients in parentheses differ from v. Zeipel's values because they contain additional 
 terms. See p. 134. 
 
128 MEMOIRS NATIONAL ACADEMY OF SCIENCES. 
 
 The remaining terms in the differential equation for ndz^ are, by eq. (143), 
 
 (1 - e cos ) 
 
 -(!- cos .) tF,'< 2 > +TW - F/') - =T /" Tf t < + i ( 
 all the terms of which are of the second order whose lowest rank is the second. They therefore 
 
 contain the factor ? 
 w* 
 
 
 To obtain ndz^ it is necessary to return to eqs. (124)-(130) and make developments for 
 terms of the second order similar to those for first order. The resulting differential equation is: 
 
 ~n&, = (1 ~ C 2 COS) {7^<') - (nte/O) } Wf > - (1 - e cos ) w 
 
 { 
 
 roi 
 
 *vr 
 
 ' - 1 - e cos 
 
 -[(l -e cos e) ^ 
 
 The sum of the last two equations, when integrated, gives the terms of second order having 
 
 m' 2 
 the factor ^ It has been shown by v. Zeipel through computation and we have shown ana- 
 
 lytically that 
 
 and 
 
 [(1 -e cos s) F/ 1 ^!^^ 1 )- (nfe t ('))} +w^(n5 2 / 2 )) =0- 
 Therefore, 
 
 = l-e cos 
 
 -[(!- cos 
 
 The integral is tabulated in Table XXXIV. 
 Summarizing, we have included first order terms in 
 
 
 given by tables XXX, XXXI, XXXII and second order terms in 
 
 
 given by Tables XXXIII and XXXIV. The addition of Tables XXX-XXXTV gives the short 
 period terms in nfe, or, the function 
 
 ndz-[ndz] 
 which is tabulated in Table XXXV. 
 
 Returning now to the differential equation for tf, the evaluation of F (#, e) and its derivatives 
 in Z 78, eq. (127) gives Z 91, eq. (146). The variable does not appear; -j is a function of t? alone 
 
 Therefore the function is of long period. The integration is one step in the determination of 
 [ndz], the long period terms in the perturbations of the mean anomaly. 
 The function [(1 -e cos e) W] is tabulated in Table XXIX6. 
 
 The function f(l - e cos i)( W- ^sV W+ ^S\], computed from Tables XXIXa and XXIXc, 
 is given in Table XXXVI. 
 
No. 3.] 
 
 MINOR PLANETS LEUSCHNER, GLANCY, LEVY. 
 
 129 
 
 The function (1 - e cos e) (0, + 0, + 0,) 
 First, 0- 
 
 is computed as follows: 
 
 5W 
 
 is given by Z 93, eq. (150) by means of Table XXXV, and -=r= is readily written by inspection 
 
 of Table XXIXa. Performing the indicated multiplications and retaining only the terms which 
 are independent of e, we have the required function as tabulated in Table XXXVII. 
 
 By eq. (146), the sum of Tables XXIX6, XXXVI, and XXXVII, multiplied by the factor 
 
 gives <?(?), tabulated in Table XXXVIII. 
 
 TABLE XXXIII. 
 
 Unit-l" 
 
 
 Sin 
 
 tc- 
 
 te 
 
 U* 
 
 u>> 
 
 I? 
 
 f+4<+4J 
 
 - 0. 316 
 
 + 1.59 
 
 -3.6 
 
 * 
 
 e+4tf+3J 
 
 + 0. 114 
 
 - 0.67 
 
 + 1.8 
 
 f 
 
 - t+2iJ+2J 
 t+2t)+2J 
 +6t>+6J 
 25+40 +4J 
 
 + 2.62 
 + 4.42 
 + 1.80 
 + 0.16 
 
 - 16.8 
 - 28.4 
 - 11.7 
 - 0.8 
 
 + 1.8 
 
 *v 
 
 - +2t>+ J 
 
 +2tJ+3J 
 t+2^+ J 
 +6>+5J 
 2s+4^+3J 
 
 - 6.18 
 - 1.90 
 - 5.57 
 - 3.95 
 - 0.06 
 
 + 36.9 
 + 13.6 
 + 32.8 
 + 23.6 
 + 0.3 
 
 - 0.9 
 
 1" 
 
 - e+20 
 J+20+2J 
 +6<>+44 
 
 + 4.04 
 + 2.12 
 + 1.90 
 
 - 21.4 
 - 14.4 
 - 10.8 
 
 
 f 
 
 - +20+ J-S 
 +60+5J-S 
 
 + 0.22 
 + 0.07 
 
 - 1.6 
 - 0.5 
 
 
 
 + (0-l> ) COS 
 
 
 
 
 5 
 
 
 
 - L265 
 
 + 6.35 
 
 -14.3 
 
 l' 
 
 + ^ 
 
 + 0.455 
 
 - 2.69 
 
 + 7.2 
 
 V 
 
 2 
 
 + 0.63 
 
 - 3.2 
 
 + 7.2 
 
 v 
 
 2+ A 
 
 - 0.23 
 
 + 1.3 
 
 - 3.6 
 
 1' 
 
 f 
 
 -23.8 
 
 +222 
 
 
 v 
 
 + 4 
 
 - + J 
 
 +72.9 
 +36.5 
 
 -569 
 
 -285 
 
 
 v 
 
 + 2J 
 
 
 -55.2 
 -87.3 
 
 +375 
 +653 
 
 
 *" 
 
 + J 
 
 +69.9 
 
 -439 
 
 
 ft 
 
 + ^+^ 
 
 
 -9.9 
 +23.1 
 
 + 77 
 -166 
 
 
 ;* -!' 
 
 f +^ 
 
 + ^ 
 
 + 5.2 
 -14.8 
 
 - 45 
 
 +112 
 
 
 
 m" 
 
 110379 22 9 
 
130 
 
 MEMOIRS NATIONAL ACADEMY OF SCIENCES. 
 
 TABLE XXXIV. 
 
 [Vol. XIV. 
 
 I noii-i 
 Unit-1" 
 
 
 Sin 
 
 .- 
 
 UJO 
 
 M 
 
 . 
 
 
 1" j-.i FiOJ ci;,i!(: t >-, !< 
 
 2;+4t?+4J 
 
 - 0.614 
 - 0.079 
 
 + 4.06 
 + 0.40 
 
 -10.3 
 
 ' 
 
 
 
 E+40+4J 
 
 - 0.74 
 + 1.74 
 + 0.31 
 + 0.45 
 
 + 3.7 
 -18.1 
 - 2.0 
 - 2.9 
 
 
 * 
 
 j-i-4,+3J 
 2 +6i?+5J 
 
 + 0.30 
 - 4.26 
 - 0.66 
 
 - 1.8 
 +32.0 
 + 3.8 
 
 
 ' 
 
 - +20+24 
 2+80+8J 
 
 -6.4 
 + 6.4 
 + 5.1 
 - 1.4 
 - 2.2 
 
 
 
 rV 
 
 P t -;- 
 
 - +20 + 4 
 +2<>+ 4 
 
 +6.5 +5J 
 2+8tf+7J 
 
 +12.0 
 - 0.9 
 - 8.5 
 -11.8 
 + 3.4 
 - 0.8 
 + 6.5 
 
 
 
 f 
 
 + 2J 
 
 ^-2t)+2J 
 
 +6iJ+44 
 
 - 5.1 
 + 1.3 
 + 6.4 
 
 
 
 * 
 
 - +20+ J-S 
 s-{~6t?~l~5^ S 
 2f -(-4^ -|-4d 
 
 - 0.3 
 + 0. 3 ' 
 
 + 1.4 
 
 
 
 
 +(tf-l> ) COB 
 
 
 
 j 
 
 ' 
 
 
 
 2+2i?+2J 
 
 - 1.02 
 - 0.78 
 + 0.41 
 
 - 8.4 
 + 6.0 
 - 2.5 
 
 
 '' 
 
 + 4 
 
 2t+2tf+3J 
 
 -3.25 
 + 0.58 
 - 0.31 
 
 +30.1 
 - 4.8 
 + 2.1 
 
 
 '* 
 
 _ +2i?+2J 
 
 2 
 
 2 +4<>+4J 
 
 + 3.6 
 + 1.1 
 + 0.5 
 - 0.8 
 
 :'_' 
 
 
 "' 
 
 - +20+ A 
 2+ A 
 
 - 3.4 
 + 1.6 
 
 L f . 
 
 
 , 
 
 t 
 
 - 0.36 
 
 + 2.6 
 
 
 '' 
 
 + 4 
 
 + 0.27 
 
 - 2.1 
 
 
 1 
 
 * 
 
was.] MINOR PLANETS LEUSCHNER, GLANCY, LEVY. 
 
 TABLE XXXV. 
 Logarithmic. niz-[ntz] 
 
 131 
 
 Unlt-1". 
 
 
 Sin 
 
 KT 
 
 r 
 
 vr* 
 
 
 
 V 
 
 w> 
 
 f 
 
 -+ * 
 
 
 
 
 4.1570 
 
 4.8741, 
 
 
 
 + tf + 4 
 
 
 
 
 2.7684, 
 
 3.3827 
 
 3.7172, 
 
 1? 
 
 + >+ J 
 
 
 
 
 4.0056, 
 
 4.7686 
 
 
 V 
 
 + + 4 
 
 
 
 
 4.0766, 
 
 4.8295 
 
 
 f 
 
 i + + ^ 
 
 
 
 
 4.1365 
 
 4.8738, 
 
 
 n* 
 
 + tf + 2J 
 
 
 
 
 3.3345 
 
 4.5162, 
 
 
 rf 
 
 E +30+24 
 
 
 
 
 4.2240, 
 
 4.9611 
 
 5.6685, 
 
 1 
 
 +3i>+3J 
 
 
 
 
 4.0671 
 
 4.8483, 
 
 5.5636 
 
 5" 
 
 1 J+W+3J 
 
 
 
 
 5.0926, 
 
 6.0018 
 
 
 ^ 
 
 < +W+4J 
 
 
 
 
 5. 2325 
 
 6. 1714, 
 
 
 V 
 
 1 +5i>+54 
 
 
 
 
 4.7675, 
 
 5.7344 
 
 
 J 1 
 
 .E + W+4J-J 
 
 
 
 
 3.8050, 
 
 4.7998 
 
 
 ^ 
 
 -Jf+ 1> 
 
 
 
 
 3.3112 
 
 3.8350, 
 
 4.1355 
 
 ? 
 
 -i-e+ J+ J 
 
 
 
 
 3.2065, 
 
 3. 7910 
 
 4.0833, 
 
 * 
 
 -,+30+ J 
 
 
 
 
 3.5338 
 
 4.6236, 
 
 
 *v 
 
 -< +30+24 
 
 
 
 
 4.0879 
 
 5.0382 
 
 
 ^ 
 
 +30+34 
 
 
 
 
 a 6012, 
 
 4.5318, 
 
 
 j 7 
 
 -J +30+2J-J 
 
 
 
 
 3.2074 
 
 4.1925, 
 
 
 ^ 
 
 1 
 
 
 9.868, 
 
 0.5689 
 
 2.922 
 
 3.4600, 
 
 3.3670 
 
 ^ 
 
 + ^ 
 
 
 9.482 
 
 0.2533, 
 
 2.673, 
 
 3.2959 
 
 3.1772, 
 
 it* 
 
 +20+ J 
 
 0.746, 
 
 [1.384] 
 
 3. 2927, 
 
 [4. 14906] 
 
 [4. 6990,] 
 
 
 
 +20+24 
 
 
 9.788, 
 
 2.47560 
 
 3. 10847, 
 
 3.4540 
 
 [3. 3960,] 
 
 >?' 
 
 +20+24 
 
 0.645 
 
 U-342,] 
 
 2.305, 
 
 [3. 6179,] 
 
 [4. 4018] 
 
 
 v 1 
 
 +20+24 
 
 0.326 
 
 1. 119, 
 
 2.935, 
 
 3. 3017, 
 
 [4. 39206] 
 
 
 j* 
 
 +20+24 
 
 
 
 3.4276, 
 
 4.23764 
 
 4.76933, 
 
 
 :r* 
 
 +20+34 
 +40+24 
 
 0.28, 
 
 1.102 
 
 3.1738 
 3.6004 
 
 [3. 5449,] 
 4. 27485 
 
 [3.8446,] 
 
 
 A 
 
 +40+34 
 +40+34 
 
 9.057 
 
 0.692, 
 
 3. 10161 
 4. 0519, 
 
 3.9302, 
 3.7975 
 
 4.52415 
 
 [4. 78162,] 
 
 < 
 
 +40+34 
 
 
 
 4.1385 n 
 
 4.6961 
 
 
 
 >* v 
 
 +40+34 
 
 
 
 4. 2431, 
 
 5.1290 
 
 
 
 ij 
 
 +40+44 
 
 9.500, 
 
 0.522 
 
 2.9351, 
 
 3.8035 
 
 4. 41616, 
 
 4.63017 
 
 q 3 
 
 +40+44 
 
 
 
 3. 7714 
 
 4.2108, 
 
 
 
 ^** 
 
 +40+44 
 
 
 
 4.4165 
 
 5.0931, 
 
 
 
 A 
 
 +40+44 
 
 
 
 4.1524 
 
 5.0661, 
 
 
 
 iV 
 
 +40+54 
 
 
 
 4.0588, 
 
 4.8136 
 
 
 
 j*l +40+34 -J 
 
 
 
 3.2322, 
 
 4.2342 
 
 
 
 ; ^' +40+44 -.T 
 
 
 
 2.744, 
 
 [3. 0%2] 
 
 
 
 5" +60+44 
 
 0.28 
 
 0.64 n ] 
 
 3.8027 
 
 4.77998, 
 
 5. 52852] 
 
 
 i) ij' +60+54 
 ?* +60+64 
 
 j E + 60 + 54-.T 
 
 0.596, 
 0.255 
 
 8.8 
 
 1.070] 
 0-8 n ] 
 9-3,] 
 
 3.9374, 
 3.4684 
 2.415 
 
 [4.94342] 
 [4. 50125,] 
 3.48a,T 
 
 5. 70347,] 
 5. 27451] 
 4. 2931] 
 
 
 5" +80+54 
 
 
 
 4.5564 
 
 5.4999, 
 
 
 
 i) i? 77 +80+64 
 
 
 
 4.8668, 
 
 5.8416 
 
 
 
 T,V +80 + 74 
 
 
 
 4.6990 
 
 5.7030, 
 
 
 
 if +80+84 
 
 
 
 4.0531, 
 
 5.0844 
 
 
 
 j 2 ij' +8i>+6J-^ 
 
 
 
 3.5829 
 
 4.6352, 
 
 
 
 fr, 
 
 t+80+74-2 1 
 
 
 
 3.3768, 
 
 4.4540 
 
 > ' 1 ", '-, 
 
 V- 
 
 $ 
 
 V 
 
 - +20 
 
 - +20+ 4 
 
 - +20+24 
 
 0.606 
 0.791. 
 0.418 
 
 fl.422,] 
 1. 690] 
 [1.365,] 
 
 3.2132 
 3. 3777, 
 2.894 
 
 3.6657, 
 3.8866 
 [3. 4616,] 
 
 19260 
 4.72168 
 3.8078 
 
 
 p 
 
 - + 20+ J-J 
 
 9.34 
 
 0.28 J 
 
 2.938 
 
 3. 4714, 
 
 3.7862 
 
 
 1* 
 
 - +40+ 4 
 
 
 
 3.5208 
 
 4.07255 
 
 
 
 < 
 
 - +40+24 
 
 
 
 3.4965, 
 
 4.59582 
 
 
 
 -/v 
 
 - +40+34 
 
 
 
 3. 2416 
 
 4.5467, 
 
 
 
 s 3 
 
 - +40+44 
 
 
 
 2.430, 
 
 3.9848 
 
 
 
 j 1 ?' 
 
 - f +40+24 -I 
 
 
 
 3.5496 
 
 4.19852, 
 
 
 
 /*! 
 
 - +40+34 -I 
 
 
 
 3. 3247, 
 
 4.05994 
 
 
 
132 
 
 Logarithmic. 
 
 MEMOIRS NATIONAL ACADEMY OF SCIENCES. 
 
 TABLE XXXV Continued, 
 niz [noz] 
 
 [Vol. XIV. 
 
 Unlt-l" 
 
 
 Sin 
 
 w - 
 
 ^ 
 
 ^ 
 
 u- 
 
 w 
 
 
 
 , 
 
 *e+3t?+2J 
 
 
 
 
 3. 6731 
 
 4. 0029 n 
 
 
 
 f-)-3''-i-3^ 
 
 
 
 
 2. 3528 
 
 3. 2475 B 
 
 3.9005 
 
 7;' 
 
 $+30+3J 
 
 
 
 
 3. 6181 n 
 
 4. 2122 
 
 
 J 3 
 
 fs+Stf-j-SJ 
 
 
 
 J 
 
 3. 4072 B 
 
 4.4000 
 
 
 1) y' 
 
 ^+3d+4J 
 
 
 
 
 3. 5244 
 
 4. 4012 B 
 
 
 TI' 
 
 fe+5i>+4J 
 
 
 
 
 3. 3533 
 
 4. 4231 n 
 
 5. 2725 
 
 rj 
 
 |-i-5t-t-5J 
 
 
 
 
 3. 1780 n 
 
 4. 2730 
 
 5. 1359 B 
 
 Tj' 3 
 
 fs-j-7*-(-5J 
 
 
 
 
 4. 2775 
 
 5. 4708 B 
 
 
 IT 
 
 i-j-7i>+6^ 
 
 
 
 
 4. 4051 B 
 
 5. 6177 
 
 
 
 fj+7<?+7J 
 
 
 
 
 3. 9296 
 
 5. 1605 B 
 
 
 , 
 
 2e+2tf+2J 
 
 
 9.486 
 
 2. 1744 B 
 
 2.708 
 
 [2. 889 n ] 
 
 2. 599 n 
 
 rf 
 
 2j-(-2i>+34 
 
 
 
 
 1. 946 n 
 
 2.501 
 
 2. 516 B 
 
 M! 
 
 2IK+4J 
 
 8.8 B 
 
 [0. 561] 
 8.90 B 
 
 2. 789 B 
 9.599 
 
 [3. 5813] 
 1.711 
 
 [4. 1074 B ] 
 2. 5795 B 
 
 3. 1726 
 
 T 
 
 2+4,+4J 
 
 9.2 
 
 [0. 34 n ] 
 9. 819 n 
 
 2.618 
 0. 5840 
 
 [3. 4962 B ] 
 2. 7821 
 
 [4. 0890] 
 
 4. 51865 
 
 1 
 
 2f+6t+6J 
 
 
 9.653 
 
 0. 4645 B 
 
 2. 5979 n 
 
 3' 6265 
 
 4. 38424 B 
 
 
 | -(-5^+5J 
 
 
 
 
 1.2340 
 
 2. 1166 B 
 
 2. 7076 
 
 i)' 
 
 4-|-7#+6J 
 
 
 
 
 2. 3679 
 
 3. 3518 B 
 
 4. 0587 
 
 >) 
 
 |+7t?+7J 
 
 
 
 
 2. 1758 B 
 
 3. 1926 
 
 3.9204, 
 
 
 (l>-l> ) COS 
 
 
 
 
 
 
 
 , 
 
 E 
 
 0. 1021n 
 
 0.728 
 
 2. 8978 B 
 
 3.4504 
 
 3. 7168 B 
 
 
 I 3 
 
 e 
 
 1. 377 B 
 
 [2. 346] 
 
 3. 8211 
 
 4. 6762 
 
 
 
 jj 7) /a 
 
 5 
 
 1.941 B 
 
 2.815 
 
 4. 4076 B 
 
 5. 1971 
 
 
 
 frj 
 
 e 
 
 1.364 
 
 2. 220 B 
 
 4.4076 
 
 5. 1971 B 
 
 5. 7086 
 
 
 rf 
 
 + J 
 
 9.658 
 
 0. 774 B 
 
 2. 7836 
 
 3. 3840 n 
 
 3. 6946 
 
 
 7) 3 7)' 
 
 + J 
 
 1.863 
 
 2. 755 B 
 
 4.2546 
 
 5. 0814 B 
 
 
 
 ^ /3 
 
 t+ ^ 
 
 1.844 
 
 2.642 n 
 
 4. 1953 
 
 4. 9770 B 
 
 
 
 j" 5' 
 
 + ^ 
 
 1. 170 B 
 
 2.049 
 
 4. 3715 n 
 
 5. 1770 
 
 5. 6975 B 
 
 
 I$i 
 
 + 2J 
 
 L742, 
 
 2. -574 
 
 4. 0203 B 
 
 4.8466 
 
 
 
 / >)' 
 
 *+ ^ 
 
 0.716 
 
 1.65 B 
 
 4.0809 
 
 4. 8829 n 
 
 5.4008 
 
 
 fl 
 
 
 1.00 B 
 
 1.89 
 
 4. 3427 B 
 
 5. 0837 
 
 5. 5553 B 
 
 
 ,V 
 
 -t+ ^ 
 
 1.562 
 
 2. 455 B 
 
 3.9535 
 
 4. 7803 B 
 
 
 
 ?v 
 
 2 
 2.+ J 
 
 9.801 
 9.357 B 
 
 [0. 43 n ] 
 [0.473] 
 
 2.5842 
 2. 4548 n 
 
 3. 1493 B 
 3. 0830 
 
 3. 4158 
 
 
 
 ^'^ e - 1. :| 
 
 
 9.56, 
 
 0.42 
 
 
 
 
 V 
 
 !+ j 
 
 
 9.43 
 
 0.32 n 
 
 
 
 
 1 sn A.Tg.+(#-d )2w*TiPi)'<lj 1 tC. 1 coa 
 where C lt C 3 , C 3 , represent the respective coefficients. 
 
 3 sin Arg. 
 
 Mil 
 
No. 3.) 
 
 MINOR PLANETS LEUSCHNER, GLANCY, LEVY. 
 
 TABLE XXXVI. 
 -![(! -t coe ) (W- J- 2 ) (W+J- H)] 
 
 133 
 
 Unit 4th decimal of a radian. 
 
 
 Cos 
 
 K-< 
 
 tc-> 
 
 to- 
 
 to- 1 
 
 w> 
 
 w 
 
 
 - 
 
 
 +0. 000032 
 
 -0.0080 
 
 +0. 0493 
 
 - 0. 176 
 
 +0.52 
 
 7, J 
 
 
 -0. 00028 
 
 +0. 0037 
 
 -0. 133 
 
 +L10 
 
 - 8.8 
 
 
 V 
 
 
 -0.00014 
 
 +0. 0026 
 
 -0. 095 
 
 +1.27 
 
 -14.4 
 
 
 ? ' 
 
 
 
 -0.0003 
 
 +0. 139 
 
 -1.20 
 
 + 5.9 
 
 
 iV 
 
 J 
 
 +0. 00047 
 
 -0. 0070 
 
 +0. 252 
 
 -2.51 
 
 +22.8 
 
 
 / V 
 
 n 
 
 20+24 
 
 +0. 000017 
 
 -0.00042 
 
 +0. 0437 
 
 -0. 366 
 
 + 2.10 
 
 
 'v 
 
 20+ A 
 
 -0.000006 
 
 +0.00045 
 
 -0. 0639 
 
 +0.508 
 
 - 2.79 
 
 
 / 
 
 40+44 
 
 +0.00004 
 
 +0.0006 
 
 -0.194 
 
 +1.64 
 
 -11.4 
 
 
 V 
 
 40+34 
 
 -0. 00012 
 
 -0. 0012 
 
 +0. 372 
 
 -3.59 
 
 +32.2 
 
 
 *> 
 
 40+24 
 
 +0. 00011 
 
 +0.0003 
 
 -0. 252 
 
 +2.40 
 
 -19.8 
 
 
 f 
 
 40+34-2 
 
 +0.00001 
 
 -0.0001 
 
 +0. 032 
 
 -0.19 
 
 
 
 
 +(0-0 ) sin 
 
 
 
 
 
 
 
 *V 
 
 4 
 
 
 -0.00004 
 
 
 +0.010 
 
 - 0.08 
 
 
 Ji 
 
 20+24 
 
 +0. 000066 
 
 -0.00060 
 
 +0. 0399 
 
 -0. 275 
 
 + 0.94 
 
 
 v 
 
 20+ A 
 
 -0. 000024 
 
 +0. 00047 
 
 -0. 0296 
 
 +0. 221 
 
 - 0.81 
 
 
 / 
 
 40+44 
 
 -0. 00023 
 
 +0. 0028 
 
 -0. 114 
 
 +1.02 
 
 -4.7 
 
 
 iV 
 
 40+34 
 
 +0. 00039 
 
 -0. 0053 
 
 +0. 251 
 
 -2.20 
 
 + 9.9 
 
 
 !> 
 
 40+24 
 
 -0. 00011 
 
 +0. 0024 
 
 -0. 124 
 
 +1.11 
 
 - 5.1 
 
 
 
 (0-0 ) a coe 
 
 
 
 
 
 I- t.. 
 
 
 7)' 
 
 
 -0. 00017 
 
 +0. 0014 
 
 -0. 052 
 
 +0.38 
 
 - 1.4 
 
 
 |V 
 
 A 
 
 +0. 00019 
 
 -0. 0021 
 
 +0. 077 
 
 -0.61 
 
 + 2.4 
 
 
 *" 
 
 
 -0.00005 
 
 +0.0008 
 
 -0. 029 
 
 +0.24 
 
 - 1.0 
 
 
 
 
 m 
 
 m' 3 
 
 m", m' 2 
 
 m r ' 
 
 TO" 
 
 m 
 
 TABLE XXXVII. 
 [ (00) (I ecoee)-gj- 
 
 Unit 4th decimal of a radian. 
 
 
 Cos 
 
 ., 
 
 -, 
 
 to- 
 
 ~ 
 
 J 
 
 w 
 
 , 
 
 
 
 
 +0. 000042 
 
 -0. 01071 
 
 + 0.0883 
 
 - 0. 402 
 
 + 1.31 
 
 - 3.9 
 
 9 1 
 
 
 -0. 00043 
 
 +0. 0056 
 
 -0. 189 
 
 + 2.73 
 
 - 51.3 
 
 + 299 
 
 
 1 ! /2 
 
 
 -0. 00021 
 
 +0. 0048 
 
 -0. 296 
 
 + 4.47 
 
 - 59.8 
 
 + 416 
 
 
 ? 
 
 
 
 -0.0004 
 
 +0. 186 
 
 -2.00 
 
 + 11.7 
 
 - 40 
 
 
 Tl 71 
 
 J 
 
 +0. 00076 
 
 -0. 0110 
 
 +0. 530 
 
 - 7.59 
 
 +104.2 
 
 - 682 
 
 
 Tl 
 
 20+24 
 
 +0. 000055 
 
 -0. 00086 
 
 +0. 1005 
 
 - 1.153 
 
 + 21. 86 
 
 - 81.5 
 
 +217 
 
 If' 
 
 20+ 4 
 
 -0. 000020 
 
 +0.00090 
 
 -0. 1377 
 
 + 1.463 
 
 -9.50 
 
 + 44.2 
 
 -176 
 
 Tj 3 
 
 40+44 
 
 -0.00031 
 
 +0. 0041 
 
 -0. 477 
 
 + 6.49 
 
 -133. 8 
 
 + 708 
 
 
 J) 1)' 
 
 . 40+34 
 
 +0. 00068 
 
 -0. 0084 
 
 +1. 295 
 
 -17.43 
 
 +261. 3 
 
 -1266 
 
 
 " 
 
 40+24 
 
 -0. 00030 
 
 +0. 0041 
 
 -0. 921 
 
 +11. 58 
 
 - 95.2 
 
 + 452 
 
 
 ' 
 
 40+34 -S 
 
 -0.00001 
 
 +0.0001 
 
 -0. 036 
 
 + 0.58 
 
 -5.3 
 
 + 25 
 
 
 TJ 7} 
 
 4 
 
 0.00000 
 
 -0.0004 
 
 +0. 052 
 
 - 0.44 
 
 + 2.0 
 
 
 
 TI 
 
 20+24 
 
 +0. 000044 
 
 -0. 00052 
 
 +0. 0266 
 
 - 0.212 
 
 + 0.83 
 
 
 
 Jj' 
 
 20+ 4 
 
 -0. 000016 
 
 +0. 00038 
 
 -0. 0197 
 
 + 0. 170 
 
 - 0.70 
 
 
 
 r 3 
 
 40+44 
 
 -0. 00031 
 
 +0. 0037 
 
 -0. 153 
 
 + 1.62 
 
 -7.9 
 
 
 
 1j Jj' 
 
 40+34 
 
 +0. 00052 
 
 -0. 0072 
 
 +0. 335 
 
 - 3.40 
 
 + 16.9 
 
 - .7.77, 
 
 !'! 
 
 1* 
 
 40+24 
 
 -0. 00015 
 
 +0. 0032 
 
 -0. 165 
 
 + 1.68 
 
 -8.6 
 
 
 
 
 
 m' 2 
 
 TO' 3 
 
 m", m' 2 
 
 TO' 2 
 
 m' 2 , m' 
 
 m'*, TO' 
 
 m' 2 , TO' 
 
134 
 
 Logarithmic. 
 
 MEMOIES NATIONAL ACADEMY OF SCIENCES. [VOLXIV. 
 
 TABLE XXXVIII. 
 
 0(0) Unit- 1 radian. 
 
 
 Cos 
 
 w- 
 
 te- 
 
 tO-4 
 
 w- 
 
 U)-l 
 
 w-i 
 
 w 
 
 w 
 
 w 
 
 
 
 
 
 1.5 
 
 [3. 909] 
 
 4.960 
 
 6. 6748 B 
 
 7. 2764 
 
 7. 540 B 
 
 7.31 
 
 'V 
 
 
 
 2.0 
 1.9 
 
 [4. 644 n ] 
 3.41] 
 
 [5. 160] 
 4. 75 n 
 
 6. 150] 
 6. 509] 
 
 8. 048,,] 
 8. 2077 B ] 
 
 8.838 
 [8. 994] 
 
 8. 655 B 
 8. 919 B 
 
 8. 100 B 
 
 ? 
 
 ^ .M - 
 
 
 
 2.83 n 
 
 5.146 
 
 6. 299 n ] 
 
 7. 994] 
 
 8. 740 
 
 [8 r 656] 
 
 
 1* 
 
 J 
 
 
 2.34 
 
 [4. 446] 
 
 [4.57] 
 
 6. 728 B ] 
 
 8. 4022] 
 
 9. 1999 B 
 
 9.0854 
 
 8.079 
 
 1)1'* 
 
 20 
 
 1.6 
 
 [2. 6 n ] 
 
 5.744 
 
 6. 535 n 
 
 8. 3811 
 
 9. 1031n 
 
 9. 0128 
 
 
 
 ,' 
 
 20+ 4 
 
 
 0.8 n J 
 
 [3. 068.] 
 
 [5. 2988] 
 
 7. 2212 B 
 
 [7. 3772] 
 
 [8. 0372] 
 
 [8. 764 B ] 
 
 8.668 
 
 * 
 
 20+ 4 
 
 2.32 B 
 
 3.30 
 
 5. 886 n 
 
 6.718 
 
 8. 5059 ra 
 
 9. 2804 
 
 9. 201 7 B 
 
 
 
 V" 
 
 20+ 4 
 
 
 
 5. 301 n 
 
 6.149 
 
 8, 2302 B 
 
 9.0154 
 
 8. 938 B 
 
 
 
 P (' 
 
 20+ 4 
 
 
 
 
 
 8. 5592 
 
 9. 3245 B 
 
 9. 2428 
 
 
 
 f" 
 
 20+24 
 
 2.48 
 
 3.40 
 
 5.422 
 
 6. 292 n 
 
 7.476 
 
 8. 664 n 
 
 8.636 
 
 
 
 1) 
 
 20+24 
 
 
 1.22 
 
 [2. 94] 
 
 [5. 1206 n ] 
 
 7. 6416 
 
 [7. 9638 B ] 
 
 [7. 083 B ] 
 
 [8. 645] 
 
 8. 582 B 
 
 ,," 
 
 20+24 
 
 1.9 
 
 [3. B ] 
 
 5.442 
 
 6. 328 n 
 
 8. 0915 B 
 
 8. 630 B 
 
 8.742 
 
 
 
 ft 
 
 20+24 
 
 
 
 
 
 8. 5904 n 
 
 9. 3489 
 
 9. 8024 n 
 
 9. 6532 
 
 
 ,Y 
 
 20+34 
 
 2.04 n 
 
 3.00 
 
 4.98 n 
 
 5.89 
 
 8. 0326 
 
 8. 1973 B 
 
 7.69 
 
 
 
 X 
 
 20+ 4-2 1 
 
 
 
 4.51 
 
 5.42 n 
 
 8. 1011 
 
 8. 873 B 
 
 8.792 
 
 
 
 ? *' 
 
 20+24 -2 1 
 
 
 
 4.04 
 
 5.00 
 
 6.89 B 
 
 8.182 
 
 8. 158 B 
 
 
 
 ? 
 
 / 
 
 40+24 
 40+34 
 40+44 
 40+34 -2 1 
 
 
 [2. 66 n ] 
 [2. 72] 
 [2.20] 
 1. 5 n 
 
 2.7] 
 4. 369] 
 4. 624] 
 2.45 
 
 6. 1031 
 
 6. 2526 n 
 [5. 824] 
 4.68 
 
 [8. 4188] 
 8. 5594 
 [8. 0924 B ] 
 7. 1747 B 
 
 [8. 5297] 
 8. 7988 n ] 
 [8. 4338] 
 7. 301 
 
 6.0] 
 7. 94 B ] 
 7.24] 
 8.111 
 
 7.90 n 
 8.287 
 7.74 B 
 8. 127. 
 
 8.210 
 8.044, 
 
 V 3 
 
 60+34 
 
 
 
 5. 30^ 
 
 6.149 
 
 9. 1294 B 
 
 9. 7728 
 
 9. 6609 n 
 
 
 
 ?r" 
 
 60+44 
 
 
 
 5.92 
 
 6.74 n 
 
 9. 4432 
 
 0. 14644,, 
 
 0. 05077 
 
 
 
 ,v 
 
 60+54 
 
 2. O n 
 
 3.0 
 
 5.93 n 
 
 6.79 
 
 9. 2774 n 
 
 0. 03298 
 
 9. 9494 B 
 
 
 
 ? 
 
 60+64 
 
 2.0 
 
 3.0 n 
 
 5.420 
 
 6. 292 n 
 
 8.634 
 
 9. 4351 B 
 
 9. 3608 
 
 
 
 ?V 
 
 60+44 - 
 
 
 
 4.04 B 
 
 5.00 
 
 8. 272 B 
 
 9. 1028 
 
 9. 0334 B 
 
 
 
 h 
 
 60+54-JT 
 
 
 
 4.51 
 
 5.42 n 
 
 8.0554 
 
 8. 926 B 
 
 8.864 
 
 
 
 
 (0-0 ) sin 
 
 
 
 
 
 
 
 
 
 
 *v 
 
 4 
 
 
 
 [2. 60 B ] 
 
 4.71 
 
 5.94 n 
 
 6.507 B 
 
 6.606 
 
 
 
 I/ 
 
 20+ 4 
 
 
 1.36 
 
 [2. 48] 
 
 4.49 
 
 [5. 255 B ] 
 
 [5. 51] 
 
 5.25 n 
 
 
 
 7 
 
 20+24 
 
 
 1.82,, 
 
 [2.42] 
 
 4.64 n 
 
 5.350 
 
 [5. 51.] 
 
 [5. 16] 
 
 
 
 ^ /3 
 
 40+24 
 
 
 2.34 
 
 [3.00] 
 
 5.392 
 
 6. 179.] 
 
 6. 528 B 
 
 6.665 
 
 
 
 v 
 
 40+34 
 
 
 2.89 n 
 
 [3. 46] 
 
 5. 702 n 
 
 6. 467] 
 
 6.851 
 
 6. 979 B 
 
 
 
 >5 
 
 40+44 
 
 
 2.66 
 
 [3. 459] 
 
 [5. 357] 
 
 6. 127,] 
 
 6. 530 B 
 
 6.653 
 
 
 
 
 (0-0 ) 2 cos 
 
 
 
 
 ft<WO ,' 
 
 
 
 
 
 
 I} 2 
 
 m- ' 
 
 
 
 2.08 
 
 2.08 
 
 , y-l. l 
 
 5. 546 n 
 
 5.546 
 
 
 
 V 
 
 L'*i ;i - 
 
 
 
 2.54 
 
 2.54 
 
 , ' /i r' 1 
 
 5. 396 B 
 
 5.396 
 
 
 
 7 
 
 4 
 
 
 
 2.5 B 
 
 2.5 
 
 
 5.776 
 
 5. 776 B 
 
 
 
 
 m' 3 
 
 m' 3 
 
 m' 3 , m'* 
 
 m' 3 , m /2 
 
 m' 2 , m' 
 
 m' 2 , m' 
 
 m /2 , m' 
 
 m n , m' 
 
 m", m' 
 
 l cos Arg. +(0-0 )2'M)*))P7) / 9; 2 <C' 2 sin Arg. 
 where C lt C 3 , C 3 , represent the respective coefficients. 
 
 cos Arg. 
 
 COMPARISON 
 
 OF TABLES. 
 
 Table XXX. With the aid of the manuscript the source of all the discrepancies indicated 
 by brackets has been traced. Coefficients in parentheses are functions of coefficients in paren- 
 theses in Table XXVII. 
 
 Table XXXI. The function was computed by the first of Z 81, eqs. (137), which is more 
 rigid than the one following it, which v. Zeipel used. Aside from the addition of omitted terms, 
 the bracketed coefficients are more accurate by reason of the errors in v. Zeipel's Table XVIII. 
 
 Table XXXII. The computation was performed according to Z 82, eqs. (139) and (140), 
 in place of eq. (141) which is less rigid. Besides the discrepancies due to the addition of omitted 
 terms, four bracketed coefficients are of opposite sign. These discrepancies may be due either 
 
NO. 8.] MINOR PLANETS LEUSCHNER, GLANCY, LEVY. 135 
 
 to a numerical error or to the number of terms included. The remaining discrepancy is due 
 to slight inaccuracies of v. Zeipel's computation. 
 
 Table XXXIII. The discrepancy in this table follows from one in Table XVIII. Third 
 degree terms in Table XVIII were not integrated because, in the aggregate, they amount to 
 very little. 
 
 Table XXXIV. Our table is more extensive. Second degree terms are, however, not 
 complete, for they do not include second degree terms in 
 
 [%] cos + [z 2 ] sin e 
 
 The discrepancies are of no importance. 
 
 The integration of eq. (146) is best performed individually for each planet. The analytical 
 developments are as follows: 
 
 The differential equation can be written 
 
 By a change of variable 
 
 , d * . .=1+0(0) 
 
 \ 2 / 
 
 Writing 
 
 we have Z 96, eq. (152), in which the last term can be neglected. 
 
 For a given planet the factors w, TJ, j* and the argument J are known constants. There- 
 fore 1 +0 (t?) can be expressed as in eq. (153), as a Fourier series of sines and cosines of mul- 
 tiples of 2#, in which the non trigonometrical term is designated by a. 
 
 Expressing eq. (153) in terms of exponentials and solving for d ( -~s [n'te] j by the expansion 
 
 of {1+0 (#)}"', and reintroducing the trigonometric functions, we have the equation 
 following eq. (153), in which the nontrigonometrical part is taken outside the brackets as a 
 common factor. The brackets in this equation do not have the special significance which 
 they have had previously. 
 
 The variables e and t> are now separate and the integration can be performed. Trans- 
 ferring the common factor to the left-hand side of the eauation. performing the integration 
 and adding 
 
 n 
 
 as the constant of integration, we have the argument expressed as a function of t> in eq. (154), 
 where is defined by eq. (155). 
 
 The reversion of the series gives # as a function of . We have by eq. (154) 
 
 where 2Cis & small quantity. Given 
 
 z = w + <*0(z), where tx is small, 
 we have, by a theorem of Lagrange, 
 
 By means of this theorem eqs. (156), (157) can be derived, where it is to be noticed that 
 (C~ 2 C + C/ ) k an approximation for ( ). In our developments we have used ( ). 
 
136 MEMOIRS NATIONAL ACADEMY OF SCIENCES. 
 
 If in Z eq. (155) we add and subtract/ -<re [n'dz']] 
 
 \2 V 
 
 1 
 
 f Am i 7? 2\ 
 
 , O 7* {*** "T~ J-'n I y 
 
 r= 2 7 ' 
 
 C i 
 
 Substituting this value of f in eq. (156), 
 
 [Vol. XIV. 
 TIDJ! Ji Oj 
 
 iifii JTTOV 
 
 
 + Series 
 
 i :nij 7 LtfHfi 'it't'T -!-:xi.t.!<? ;;<:; ! 
 
 Substituting the last equation in eq. (145), we obtain Z 98, eqs. (159), (160), and (161). In 
 
 2 
 eq. (160) the factor (s c) is an approximation for - ( ) ; in our work we have used the latter. 
 
 2 
 Since [ndz] t is the series in eq. (156) multiplied by the factor -r 
 
 Table XXXV. With the exception of the two coefficients under the heading w*, all the 
 bracketed quantities are functions of other coefficients in parentheses or brackets, or they are 
 functions of additional terms. The two coefficients excepted seem to be in disagreement through 
 some numerical error by v. Zeipel. 
 
 Table XXXVI. Since the mass factors have not been kept explicit, it may be well to remark 
 that only the zero degree term of third order has been included under the heading w 2 . 
 
 The bracketed quantities are numerous. Aside from the accumulation of discrepancies 
 already discussed, the disagreements are to be attributed, in general, to the relative extent of 
 the computations. It is found from computation that as the number of terms included in a 
 product is increased the resulting coefficient for a given argument is numerical!}' larger. For 
 the most part our values are larger than v. Zeipel's. Hence the discrepancies are explained by 
 assuming that our computation is more extensive. On the other hand, the function is com- 
 puted much more accurately than is necessary, and many of our disagreements are less important 
 than they appear to be. 
 
 Table XXXVII. The comparison of Tables XXXVII is similar to that for Tables XXXVI 
 with the exception that our values are not, in general, numerically larger. Some are larger 
 and some are smaller. Below are brief tables showing to what extent we used the necessary 
 series. The 0, 1, 2 signify the degrees of the terms included. 
 
 1-1 
 
 to-' 
 
 IP 1 
 
 w* 
 
 w 
 
 w' 
 
 w* 
 
 w o 
 
 w 
 
 to 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 
 2 
 
 2 
 
 2 
 
 
 2 
 
 2 
 
 
 
 
 
 
 
 
 
 m' 
 
 m" 
 
 in! 
 
No. 3.] 
 
 MINOR PLANETS LEUSCHNER, GLANCY, LEVY. 
 
 - coe c)W-[(l-ecoee)W]} 
 
 137 
 
 MLI 
 
 - 
 
 w- 
 
 V 
 
 V 
 
 V 
 
 w> 
 
 
 
 ii 
 
 If! 
 
 e 
 
 V 
 
 w> 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 
 
 1 
 
 1 
 
 1 
 
 2 
 
 2 
 
 2 
 
 2 
 
 2 
 
 2 
 
 
 
 
 
 
 m' 
 
 m" 
 
 m" 
 
 Table XXXVTIL All the bracketed quantities probably contain only the accumulation 
 of the discrepancies in Tables XXIX6, XXXVI, and XXXVII. This is a very important 
 table, and it is from differences in 9 (tf) that the perturbations may be expected to differ most. 
 
 PERTURBATIONS OF THE RADIUS VECTOR. 
 
 TIT 1 * 
 
 If Wand A are tabulated and the computation is performed in duplicate, it is not necessary 
 3 
 
 to make the long developments and the auxiliary tables in Z 6, 99-114. For this reason the 
 formulae in 6 have not been checked and the list of errata does not cover this section. 
 The essential formulae are given in Z 99. By Z 7, eq. (36), 
 
 In order to parallel the form of ndz, we write 
 
 where (flt + 0, + 0,) is given by Z 93, eq. (150). 
 
 Hence the computation proceeds as follows: the perturbation is computed by eq. (36), 
 the argument is replaced by #, and a corrective term which is the product of (0i+0, + s ) 
 and the derivative of the function with respect to # is added. The perturbation v is then 
 expressed as a function of #. It is tabulated in Table XTJTT. 
 
 Table XLHI. If there are no errors of calculation in the construction of the table, all the 
 discrepancies are due to the accumulation of other discrepancies previously discussed. 
 
 The perturbation v =/(0) includes 
 
 M 
 
 W-* 
 
 
 
 te 
 
 w> 
 
 - 
 
 m* 
 
 
 
 Iff! 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 2 
 
 2 
 
 2 
 
 
 2 
 
 2 
 
 
 3 
 
 3 
 
 
 
 3 
 
 3 
 
 
 m' 
 
 m' 2 
 
 where the tabulated numbers signify the degrees of the terms included and where only TF, and Ej 
 are inclusive of third degree. 
 
138 
 
 MEMOIRS NATIONAL ACADEMY OF SCIENCES. 
 
 TABLE XLIII. 
 
 Logarithmic. 
 
 [Vol. XIV. 
 
 Unit-1". 
 
 
 Cos 
 
 r 
 
 ur 
 
 HT' 
 
 w" 
 
 .w 
 
 to" 
 
 
 
 
 8.72 
 
 [9. 88 B ] 
 
 1. 6349 
 
 2. 1070 B 
 
 2. 2333 
 
 7 2 
 
 
 9.80 
 
 [0.212,] 
 
 
 2.759 
 
 3. 4922 n 
 
 
 V 
 
 
 8.9 
 
 9.23 
 
 
 2.937 
 
 3. 6295 n 
 
 
 ? 
 
 
 
 
 
 2. 937 B 
 
 3. 6295 
 
 
 t^ 
 
 J 
 
 9.66 B 
 
 9.78 
 
 
 3. 1136 B 
 
 3.8440 
 
 
 \\* 
 
 2t 
 
 0. 556 n 
 
 1.204 
 
 3. 2111 B 
 
 3. 7970 
 
 
 
 1 
 
 20+ J 
 
 
 0.504,, 
 
 2. 3472 
 
 2. 456 B 
 
 2. 686 B 
 
 3. 4735 
 
 rfi 
 
 20 + J 
 
 0.997 
 
 , 1.711 n 
 
 3. 6559 
 
 4. 3103 B 
 
 
 
 1* 
 
 20 + 4 
 
 0.438 
 
 1. 220 B 
 
 3. 3654 
 
 4. 0763 B 
 
 
 
 j 1 r,' 
 
 20 + J 
 
 
 
 3. 6975 B 
 
 4. 3810 
 
 
 
 1) 
 
 20 +2 J 
 
 
 0.438 
 
 2. 952 B 
 
 3. 2529 
 
 [3.0689 B ] 
 
 3. 3979 B 
 
 V 
 
 2J+2J 
 
 0. 732 n 
 
 1.497 
 
 3. 2410 B 
 
 4. 0643 
 
 
 
 11" 
 
 2rf+2J 
 
 0. 772 B 
 
 1.589 
 
 3. 4136 
 
 4. 0723 
 
 
 
 X 
 
 2i>+2J 
 
 
 
 3.9048 
 
 4. 5649 B ] 
 
 4.9303 
 
 J > J ' Ji f'bt '- ' 
 
 * V 
 
 2i?+3J 
 
 0.505 
 
 1.344, 
 
 3. 4757 n 
 
 2. 783] 
 
 
 
 & 
 
 2i>+ A-S 
 
 9.33 n 
 
 0.15 
 
 2. 938 B 
 
 3. 530] 
 
 
 
 3V 
 
 23+2J-2 
 
 9.20 
 
 0.10 n 
 
 2. 0251 
 
 3. 2961 B 
 
 
 
 ,;- 
 
 +2J 
 4,?+3J 
 
 8.9 
 9.75 n 
 
 1. 2819 B 
 [1. 5024] 
 
 3. 5514 
 3. 7885 B 
 
 3. 6173,, 
 4. 1394] 
 
 3. 8147 
 4. 3110 B 
 
 it'll M 
 
 / 
 
 4^+44 
 4V+3J-2 
 
 9.98 
 
 [1. 1342 B ] 
 9.64 n 
 
 3.4007 
 2.305 
 
 3. 9091 B ] 
 2. 542 B 
 
 [4. 1480] 
 [2. 749 n ] 
 
 
 ,'3 
 
 6^+34 
 
 0.436 
 
 1. 220 n 
 
 4. 2675 
 
 4. 7993 B 
 
 
 
 ft* 
 
 6^+4J 
 
 1. 125 B 
 
 1.862 
 
 4. 6479 B 
 
 5. 2324] 
 
 
 
 ?V 
 
 6tf+5J 
 
 1.198 
 
 1.947 B 
 
 4. 5397 
 
 5. 1768,] 
 
 
 & :IT 
 
 ? 3 
 
 6J+6J 
 
 0. 732 B 
 
 1.508 
 
 3. 9457 B 
 
 4. 6328] 
 
 
 
 P *' 
 
 60+4 J-2 1 
 
 9.20 
 
 0.10 B 
 
 3. 4099 
 
 4. 1710 B 
 
 
 
 V 
 
 6^+5J--T 
 
 9.70 B 
 
 0.56 
 
 3. 2601 B 
 
 [4. 0542] 
 
 
 
 >M' 
 
 ie+ <> 
 
 
 
 
 3,4878 B 
 
 4. 1106 
 
 i 
 
 
 ^+ t>+ J 
 
 
 
 8.3 B 
 
 2. 2106 
 
 2. 7179 n 
 
 2.919 
 
 ; 2 
 
 is+ t5+ J 
 
 
 
 
 3. 5709 B 
 
 4. 2261 
 
 
 % 2 
 
 ie+ ?+ 4 
 
 
 
 
 3.4507 
 
 4. 1296 n 
 
 
 V 
 
 i+ l>+ A 
 
 
 
 
 3.5100 
 
 4. 1837 B 
 
 
 tt* 
 
 }.+ iJ+24 
 
 
 
 
 2. 579 n 
 
 3. 9270 
 
 
 ?' 
 
 is+3^+2J 
 
 
 
 0.08 
 
 3. 6873 
 
 4. 1471 B 
 
 4. 7839 
 
 ^ 
 
 4+3t+3J 
 
 
 
 9.5 
 
 3. 5727 n 
 
 4. 1511 
 
 4. 7545 n 
 
 ," 
 
 is+5<J+3J 
 
 
 
 
 4. 5568 
 
 5. 1414, 
 
 
 tr 
 
 i+5<5+44 
 
 
 
 
 4. 7261n 
 
 [5. 4067] 
 
 
 / 
 
 i+5<>+5J 
 
 
 
 > a JOTIO ti< 
 
 4. 2862 
 
 [5. 0418n] 
 
 
 / 
 
 if+5<>+4J-2' 
 
 
 
 UoiUlflUil 
 
 3. 2570 
 
 4.0005 n 
 
 
 y" 
 
 -*+ <> 
 
 
 
 1. 086 B 
 
 2.7090 
 
 3. 3467 B 
 
 3. 7098 
 
 i| 
 
 -<:f+ I>+ J 
 
 
 
 0.88 
 
 2. 1967 n 
 
 3. 0952 
 
 3. 5836 B 
 
 V 2 
 
 -- U+3<?+ 4 
 
 
 
 
 2.514 
 
 4. 1049 n 
 
 
 7V 
 " 
 
 -,:J + 3tf + 2J 
 
 
 
 
 4. 0853 
 
 [3. 9122] 
 
 
 
 -i +3i5+3J 
 
 
 
 
 3. 8341 B 
 
 [3. 8U8] 
 
 
 f 
 
 -it+3iJ+2J-J 
 
 
 
 
 2.416 
 
 3. 6926 n 
 
 
 ij 
 
 e 
 
 
 9.62 
 
 0.58 B 
 
 2. 143 B 
 
 2. 682 2. 9151 n 
 
 V 
 
 + ^ 
 
 
 9.04 B 
 
 9.9 
 
 2.061 
 
 2. 666 n 2. 9477 
 
 tV 
 
 + 2l>+ J 
 
 0.444 
 
 1. 1661 B 
 
 3. 0588 
 
 3. 8035 B 
 
 [4. 2554] 
 
 
 e+2i?+2J 
 
 
 9.487 
 
 2.]744 B 
 
 2. 7280 
 
 2. 972 n 
 
 2.976 
 
 >! 2 
 
 +2tf+2J 
 
 0.344n 
 
 1.1143 
 
 2. 692 B 
 
 [3. 5334] 
 
 [4. 0772 B ] 
 
 
 I" 
 
 +2tJ+2J 
 
 0. 025 n 
 
 0.828 
 
 2.634 
 
 3. 0726 
 
 4. 0416 B 
 
 
 f 
 
 +2^+2J 
 
 
 
 3. 1265 
 
 3. 8806 B 
 
 4. 3473 
 
 
 W' 
 
 +2tf+3J 
 
 9.98 
 
 a sii. 
 
 2. 873 B 
 
 3. 1697 
 
 [3. 5856] 
 
 
 tr* 
 
 +4i?+2J 
 
 1.105 
 
 L89W 
 
 2.864 
 
 4. 3477 n 
 
 
 
 r 
 
 +4^+3J 
 
 8. 8 n 0. 398 
 
 2. 8000 n 
 
 3. 5327 
 
 4. 0065 B 
 
 4. 3207 
 
 ,Y 
 
 +4^+3J 
 
 1. 260 n 2. 083 
 
 3. 0931 
 
 4. 4160 
 
 
 
 ," 
 
 e+4<+3J 
 
 
 3. 8375 
 
 4. 0446 B 
 
 
 
 J 1 >>' 
 
 +4^+3J 
 
 0. 267 1. 15 B 
 
 3.9421 
 
 4. 6972 B 
 
 
 
 j) 
 
 e+4tf+4J 
 
 9.19 
 
 0. 248 B 
 
 [2. 6356] 
 
 3. 4317 B 
 
 3. 9469 
 
 4. 2558, 
 
 ^ 3 
 
 +4^+4J 
 
 0. 774 1. 66,, 
 
 3. 0934 B 
 
 [3. 7866 n ] 
 
 
 
 ,," 
 
 s+4i>+44 
 
 
 4. 1154 B 
 
 4. 5547 
 
 
 
 f. 
 
 +4tJ+4J 
 
 0. 455 B 1. 32 
 
 3. 8518 n 
 
 4. 6436 
 
 
 
 Jv 
 
 +4^+5J 
 
 
 3. 7579 
 
 4. 3244 n 
 
 
 
No. 3.] 
 
 Logarithmic. 
 
 MINOR PLANETS LEUSCHNER, CLANCY, LEVY. 
 
 TABLE XLIII Continued. 
 
 139 
 
 Unlt-1". 
 
 
 Cos 
 
 - 
 
 r 
 
 * 
 
 . 
 
 v> 
 
 
 fi 
 
 e+4t+3J-J 
 
 
 
 3.0030 
 
 3. 8869, 
 
 
 
 I 3 n' 
 
 , 4i , , 4j_^ 
 
 
 
 2.4425 
 
 1 85, 
 
 
 
 '/ 
 r 
 
 g-j-6(?~l~4^ 
 
 9.98, 
 
 0.480 
 
 3. 5016, 
 
 4l 3723 
 
 4. 9952, 
 
 
 
 -{-6i?-f~5J 
 +6t?+6J 
 e~}*6i?-(-5^ ^ 
 
 0.296 
 9.95, 
 8.5, 
 
 0.823, 
 0.538 
 [9. 15] 
 
 3. 6369 
 3. 1685, 
 2.114, 
 
 [4- 5582,] 
 [4. 1334] 
 3.0881 
 
 5. 2093 
 [4. 8131,] 
 [3. 7886,] 
 
 
 3 i" 
 
 t-f"8^-|~5^ 
 
 
 
 4. 2554, 
 
 4. 9349 
 
 
 
 !* 
 cr 
 
 f-j-8i?-|-7^ 
 
 1.320 
 1.228, 
 
 2. 152, 
 2.093 
 
 4.5657 
 4. 3995, 
 
 5. 3010, 
 5. 1827 
 
 
 
 ,' 
 
 P^-gjj-Lgj 
 
 0.648 
 
 1.54, 
 
 3.7543 
 
 4. 5812, 
 
 
 
 / 
 
 ^j-gjj -i- j2 
 
 
 
 3. 2818, 
 
 4.1442 
 
 
 
 A 
 
 s-\~o^-\-7^ ~~ 2 
 
 
 
 3. 0763 
 
 3. 9759, 
 
 
 
 
 
 - t+2d 
 
 0.305 
 
 PL 1007,] 
 
 2.912 
 
 3.4958, 
 
 3. 8151] 
 
 
 j/ 
 
 - ;JgJ^ 
 
 0.490, 
 0.117 
 
 [1. 3330] 
 
 2:288, 
 
 [3. 7273] 
 3. 2375,] 
 
 4. 3119 
 3. 7892 
 
 
 a 
 
 ~f"2l?-j- J 
 
 9.04 
 
 9.96, 
 
 2.636 
 
 3. 2817, 
 
 3. 6568 
 
 
 B 
 
 ~f"4i?-{- J 
 
 
 
 3. 2197 
 
 3.9650, 
 
 
 
 
 - e+4t>+2J 
 
 1. 146, 
 
 1.89 
 
 3.0204 
 
 4.2441 
 
 
 
 n 2 j/ 
 
 f~|-4i?-l-3^ 
 
 1.005 
 
 1.78, 
 
 3. 5247, 
 
 4.0012, 
 
 
 
 B 
 
 f-|-4(?-t-4i 
 
 0.290, 
 
 1.15 
 
 3. 1793 
 
 2.982 
 
 
 
 1 K 
 
 -{~4tJ-}-2J ^* 
 
 
 
 3. 2486 
 
 4.0585, 
 
 
 
 
 - f+4tf+3J-J 
 
 9.98, 
 
 0.8 
 
 2.957, 
 
 3.8580 
 
 
 
 n 
 
 | + ,j+ J 
 
 
 
 9.0 
 
 2.3363 
 
 [3.0704,] 
 
 [3. 5111] 
 
 ,; 
 
 | + 0+2J 
 
 
 
 9.5 
 
 1.500 
 
 2.3585 
 
 3. 1842, 
 
 
 |+3*+2J 
 
 
 
 
 2.779 
 
 3. 7820, 
 
 
 
 If+SiJ+SJ 
 
 
 
 9.28 
 
 2. 1614, 
 
 3. 0257 
 
 3. 6491, 
 
 i) 1 
 
 j_j_3 ( j_i_3j 
 
 
 
 
 1.32 
 
 2.966 
 
 
 V 
 
 *+3iJ+3J 
 
 
 
 
 3.3450 
 
 4. 1111, 
 
 
 j * 
 
 |+3i>+3J 
 
 
 
 
 3. 2309 
 
 4.1965, 
 
 
 
 |+3<?+4J 
 
 
 
 
 3. 2994, 
 
 4.1520 
 
 
 ' 
 
 f+5<>+4J 
 
 
 
 1.017 
 
 3. 1617, 
 
 4.1967 
 
 5.0160, 
 
 - 
 
 5+5<>-i-5J 
 
 
 
 0.88, 
 
 2.9688 
 
 4. 0380, 
 
 4. 8781 
 
 fl" 
 
 
 
 
 
 4.0855 n 
 
 5. 2422 
 
 
 5"!' 
 
 ^+7<)+6J 
 
 
 
 
 4. 1991 
 
 5. 3823, 
 
 
 
 f+7iJ+7J 
 
 
 
 
 3. 7114, 
 
 4.9188 
 
 
 f 
 
 | ^_7,j-(-6J 2" 
 
 
 
 
 2.615, 
 
 3.8317 
 
 
 
 
 
 
 
 
 
 
 y ,' 
 
 
 
 
 
 3. 2411 
 
 3. 7872, 
 
 
 .2 
 
 ' + tj+ J 
 
 
 
 
 2. 819, 
 
 3.4476 
 
 
 y 3 
 
 -*H- tf -I 1 
 
 
 
 
 2. 9181, 
 
 3. 4813 
 
 
 B 
 
 2 
 
 
 
 
 2.364, 
 
 3. 0737 
 
 
 TJ Ij 
 
 2+ J 
 
 
 
 
 2.624 
 
 3. 3489, 
 
 
 U/z 
 
 2 S + 24 
 
 
 
 
 2.207, 
 
 2.978 
 
 ' X *lt 
 
 y 2 
 
 2+ J+^ 
 
 
 
 
 2.620, 
 
 3.2765 
 
 
 ,j 
 
 2+2<?+2J 
 
 
 
 9 8, 
 
 1.63 
 
 2. 362, 
 
 2.873 
 
 r/ 
 
 2e+2i>+3J 
 
 
 
 9.5 
 
 1.796 
 
 2.303, 
 
 2.1007 
 
 
 2 +4<J+3J 
 
 
 
 1.92, 
 
 2.700 
 
 
 
 2j+4!y+4J 
 
 
 8. 7 [8. 8] 
 
 1.5802, 
 
 2.4158 
 
 2. 9867, 
 
 gl 
 
 2+4tj+4J 
 
 
 
 
 2.330 
 
 3.1764, 
 
 
 jj'2 
 
 2j+4^+4J 
 
 
 
 
 3. 1079 
 
 3.9008, 
 
 
 /* 
 
 2j+4tj+4J 
 
 
 
 Hi t):'f?>l;i|'ni 
 
 2.736 
 
 3.6809, 
 
 . . p V .,} 
 
 " . 
 
 5 T 
 
 2j+4^+5J 
 
 
 
 
 2. 9881, 
 
 3.8425 
 
 - 
 
 
 il+S+ej 
 
 
 9.64 
 
 [0. 53] 
 [0. 36*] 
 
 2.652, 
 2.4419 
 
 3.6204 
 3. 4512, 
 
 4. 3279, 
 4. 1892 
 
 l" 
 
 2+8;>+6J 
 
 
 
 
 3. 6135, 
 
 4.6784 
 
 
 n *' 
 
 2+8i>+7J 
 
 
 
 
 3. 7124 
 
 4. 8075, 
 
 ' 
 
 M 
 
 2f+8^+8J 
 
 
 .$ i'i >i 
 
 au jV* '>/J'l 
 
 3. 2109, 
 
 4. 3338 
 
 i '. in IllSilOf 
 
 y 2 
 
 2s+8d+7J-2 
 
 
 
 
 2.068, 
 
 3.2092 
 
 yxjiii jd r> 
 
 
 | c - +5t j + 5j 
 
 
 
 9.3, 
 
 1.140, 
 
 2.0056 
 
 2. 5727, 
 
 ,j' 
 
 +7i)+64 
 
 
 
 0.5, 
 
 2. 2749, 
 
 3. 2377 
 
 3.9184, 
 
 i 
 
 |+7tf+7J 
 
 
 
 0.3 
 
 2.0542 
 
 3.0565, 
 
 3. 7710 
 
 
 j+7<>+7J 
 
 
 
 8.1 
 
 0.43, 
 
 1.346 
 
 1.959, 
 
140 
 
 MEMOIRS NATIONAL ACADEMY OF SCIENCES. 
 
 TABLE XLIII Continued. 
 
 Logarithmic. 
 
 [Vol. XIV. 
 
 Unit-1" 
 
 
 Cos 
 
 ur 
 
 ur 
 
 KT' 
 
 UJO 
 
 to 
 
 . 
 
 
 (t>-t )sin 
 
 
 
 
 
 
 
 99' 
 
 ^ 
 
 9.66 
 
 0. 810 B 
 
 2. 7559 
 
 3.3840 n 
 
 3. 6946 
 
 
 9 / 
 
 2t>4- 4 
 
 
 9.79 B 
 
 0.54 
 
 
 
 
 1 
 
 20+2J 
 
 
 9.92 
 
 [0. 63] 
 
 
 
 
 1 
 
 I 
 
 9.801 B 
 
 0.425 
 
 2. 5970 n 
 
 3. 1493 
 
 3. 4158 B 
 
 
 I 3 
 
 
 
 1. 075 n 
 
 2.045 
 
 3. 5201 B 
 
 4. 3751 
 
 
 
 II' 1 
 
 e 
 
 1.640 n 
 
 2.514 
 
 4. 1066 n 
 
 4. 8961 
 
 
 
 ft 
 
 1 
 
 1.063 
 
 1. 916 n 
 
 4. 1066 
 
 4. 8961 B 
 
 5. 4076 
 
 
 1' 
 
 + 4 
 
 9.36 
 
 0. 471 B 
 
 2. 4824 
 
 3. 0830 B 
 
 3. 3936 
 
 
 1\ 
 
 + J 
 
 1.565 
 
 2. 456 n 
 
 3. 9671 
 
 4. 7890 n 
 
 
 
 
 t+ J 
 
 1.543 
 
 2. 341 B 
 
 3. 8942 
 
 4. 6760 B 
 
 
 
 ? ';, 
 
 + J 
 
 0.87 B 
 
 1.75 
 
 4. 0705 B 
 
 4. 8759 
 
 5. 3965 B 
 
 
 
 t+ 2J 
 
 1. 441 B 
 
 2.273 
 
 3. 7192 n 
 
 4.5456 
 
 
 
 f i f 
 
 t+ I 1 
 
 0.42 
 
 1.36 B 
 
 3. 7799 
 
 4. 5819 B 
 
 5. 0998 
 
 
 
 e+ ^+2 
 
 0. 695 B 
 
 1.585 
 
 4.0417 B 
 
 4. 7827 
 
 5. 2543 B 
 
 
 ,a 
 
 j+4,>+4j 
 
 
 9.59 B 
 
 0.45 
 
 
 
 
 i' 
 
 +4tf+3J 
 
 
 9.46 
 
 0.34, 
 
 
 
 
 1 
 
 2t+2*+2J 
 
 
 9.45 
 
 [ H] 
 
 
 
 
 * 
 
 2 -j-2<?-)-3J 
 
 
 9.32 B 
 
 [O^CM] 
 
 
 
 
 9V 
 
 - H- J 
 
 1.255 B 
 
 2.149 
 
 3. 6240 B 
 
 4. 4615 
 
 
 
 
 (l> 1> ) 2 COB 
 
 
 
 
 
 
 
 , 
 
 C 
 
 
 9.25 
 
 0. 117 B 
 
 
 
 
 '' 
 
 f+ J 
 
 
 9.12 B 
 
 0.02 
 
 
 
 
 
 
 TO' 2 
 
 TO' 2 
 
 m' 2 , TO' 
 
 TO' 
 
 m' 
 
 TO' 
 
 cos Arg.+(tf-i? )^u.-r / P7j / 9; 2 C 2 sin Ar 
 where CD C 2 , C a represent the respective coefficients. 
 
 PERTURBATIONS OF THE THIRD COORDINATE. 
 
 Arg. 
 
 For the third coordinate the developments are limited to perturbations of the first order 
 and of the first degree with the exception of some secular terms of second degree. We can 
 therefore use osculating elements in this section, and use 6 and # without distinction. 
 
 By Z 8 eq. (39), 41, eq. (83) and 8, eq. (41) the equations Z 115, (192) are given, in which 
 2 is defined. 
 
 Since 
 
 dS_SS SS ^ = 2 
 de de + 3d de 
 
 By Z 9, eq. (45) we have, with sufficient accuracy, Z 115, eqs. (193). Within these limits, 
 
 dO w 
 
 Substituting this relation in the above equation and in eq. (192) in turn, the differential equation 
 to be integrated is (194). 
 
 Since F, 6, H are power series in w, it is evident from eqs. (192) that 
 
 j O 
 
 ^= 
 where 
 
NO. s.] MINOR PLANETS LEUSCHNER, GLANCY, LEVY. 141 
 
 Therefore, eq. (194) becomes 
 
 Comparing the coefficients of like powers of w on either side of the equation, it is evident 
 that the integral must be of the form 
 
 Substituting this relation in the preceding equation and equating like powers of w, the system 
 of equations (195_,) (195 t ) follows. 
 
 Within the extent of the following developments one more equation should be written by 
 analogy. 
 
 dW 
 This system of equations is integrated in a manner similar to that for -5- (see p. 81). Each 
 
 equation is broken up into two equations, one a function of e and one independent of e. The 
 differential equation (194) is then replaced by eight differential equations, the integrals of which 
 can be obtained in the order, 
 
 S_,, GS.-DSJ), [SJ, (S^-DSfJ), [SJ, 
 
 As in the case of -j , the condition is imposed that 
 
 The equivalent equations are (196)-(200). 
 
 dW" 
 A comparison of the differential equations for (S< [S<]) with the expressions for , * 
 
 dW " 
 
 s- 5 leads to an analogous form of integration for certain terms. Within the extent of our 
 
 developments, 
 
 and 
 
 1 
 -(l-cos) . -c -!- cos 
 
 dW " d W " 
 take the place of ^ and ^77 respectively. Without change of notation for the third 
 
 coordinate, (S-[S]) is given by eqs. (201), (202), where P, G, Q are computed from F, G, H in 
 
 Tables XII-XTV, by means of eqs. (118) and (119). The coefficients P, G, S are tabulated in 
 Tables L to LIT. 
 
 The function [S] is obtained from the integration of eq. (203). A constant of integration 
 is added, which is the same in form as Hansen's constant of integration for the perturbation of 
 the third coordinate, namely, 
 
 c,(cos <f> e) +Cj sin <f> Z eq. (204) 
 
 where c l and c t are undetermined. 
 
 By eqs. (192), the pertubation '. is derived from 
 
 COS c 
 
 
 . 
 i cos i 
 
 The perturbation comprises the computed value of eq. (202), the trigonometric sine series given 
 by Tables L to LII (which can be written by inspection with the aid of Table XV6), the series 
 forming Table LHI, and the constant of integration (204), in all of which 
 
142 
 
 MEMOIRS NATIONAL ACADEMY OF SCIENCES. tvoi.xiv. 
 
 TABLE L. Unit 1". 
 
 
 n 
 
 
 
 l 
 
 2 
 
 3 
 
 4 
 
 5 
 
 
 J' 1 . (n+l.-n+l)-Hr / 
 
 + 52.7 
 
 + 96.0 
 
 + 57.0 
 
 + 33.8 
 
 + 20.1 
 
 + 12.0 
 
 
 *io(** 1. 1+1)+^' 
 
 +158. 2 
 
 
 - 285.0 
 
 -101.4 
 
 - 47.0 
 
 - 24.0 
 
 
 ?Vo( n +l- n 1) ' 
 
 -158. 2 
 
 -191. 9 
 
 - 95.0 
 
 -50.7 
 
 - 28.2 
 
 - 16.0 
 
 
 *Vo(-l.--l)-*' 
 
 - 52.7 
 
 -191. 9 
 
 - 285.0 
 
 
 + 140. 9 
 
 + 48.1 
 
 b 
 
 J*,. (n+l.-n+l)+^ / 
 
 -201 
 
 -352 
 
 - 253 
 
 -176 
 
 - 119 
 
 - 80 
 
 M 
 
 J" 1 . (n-l.-n+l)+^ 
 
 -812 
 
 
 +1495 
 
 +594 
 
 + 305 
 
 +172 
 
 5 
 
 /,. (n+l.n I);:' 
 
 +812 
 
 +897 
 
 + 498 
 
 +297 
 
 + 183 
 
 +114 
 
 ( 
 
 JWn-l.-n-l)-,:' 
 
 +201 
 
 +513 
 
 + 355 
 
 
 -1478 
 
 -439 
 
 TABLE LI. 
 
 Unlt-l". 
 
 
 <3 . (n.-n+l)+K' 
 
 26.37 
 
 - 47. 98 
 
 - 28.50 
 
 16.91 
 
 10.06 
 
 - 6.02 
 
 
 Go^n.-n-l)-^' 
 
 + 79. 10 
 
 + 95. 96 
 
 + 47.50 
 
 + 25. 36 
 
 + 14. 09 
 
 + 8.02 
 
 
 5 1 . (n+l.-n+l)+T / 
 
 + 90.3 
 
 + 112. 3 
 
 + 58.5 
 
 + 29.0 
 
 + 13.6 
 
 + 5.8 
 
 
 <5,. (7l 1. 71+1)+^' 
 
 + 530. 8 
 
 + 720. 8 
 
 + 468.9 
 
 + 311. 7 
 
 + 207. 7 
 
 + 138.0 
 
 
 G,. (n+l. n 1) IT' 
 
 - 124.2 
 
 - 120. 5 
 
 - 53.3 
 
 - 21.8 
 
 - 7.4 
 
 - 1.2 
 
 
 G,. (n 1. n 1) T/ 
 
 + 609. 9 
 
 
 - 1549.1 
 
 - 674.1 
 
 - 369. 6 
 
 - 219.0 
 
 
 G . 1 (n.-n+2)+r / 
 
 - 162.4 
 
 - 211. 6 
 
 - 103. 8 
 
 - 47.7 
 
 - 19.7 
 
 - 6.4 
 
 
 <2o-i( n - n)4V 
 
 - 166.5 
 
 - 352. 6 
 
 - 298. 2 
 
 - 229.0 
 
 - 167.2 
 
 - 118.3 
 
 
 <?.,(. n) it' 
 
 + 166.5 
 
 + 96.7 
 
 + 13.2 
 
 - 14.4 
 
 - 20.7 
 
 - 19.3 
 
 
 G .i(n.-n-2)-7: / 
 
 
 + 1825.5 
 
 + 881. 4 
 
 + 516. 6 
 
 + 321. 2 
 
 + 204. 1 
 
 
 G . (n.-n+l)+ff' 
 
 + 100.4 
 
 + 176.3 
 
 + 126. 8 
 
 + 87.8 
 
 + 59.6 
 
 + 39.9 
 
 
 Go-ofa- 1) *' 
 
 - 406. 6 
 
 - 448. 6 
 
 - 249. 2 
 
 - 148.6 
 
 - 91.5 
 
 - 57.2 
 
 
 G,. (n+l.-n+l)+w / 
 
 - 432 
 
 - 592 
 
 - 370 
 
 - 218 
 
 - 122 
 
 - 64 
 
 
 ^ i -o(. n ~ 1 ~ n H~ 1 ) +f' 
 
 -2047 
 
 - 3183 
 
 - 2412 
 
 -1811 
 
 -1342 
 
 - 982 
 
 
 
 Gi.o(n+l. -n-\)x' 
 
 + 718 
 
 + 821 
 
 + 440 
 
 + 225 
 
 + 107 
 
 + 44 
 
 
 o 
 
 Gi. (nl.nl)7^ 
 
 -2401 
 
 
 +12134 
 
 +4939 
 
 +2788 
 
 +1744 
 
 1 
 
 G . l (n.-n+2)+x' 
 
 + 693 
 
 + 951 
 
 + 568 
 
 + 314 
 
 + 158 
 
 + 68 
 
 
 Gjo.^n. n)+7r' 
 
 + 893 
 
 + 1773 
 
 + 1607 
 
 +1356 
 
 +1089 
 
 + 844 
 
 
 G .j(7i. n) w 7 
 
 - 893 
 
 - 747 
 
 - 254 
 
 - 27 
 
 + 68 
 
 + 98 
 
 
 gj / _ n _9'i_ / 
 
 
 -13263 
 
 - 5889 
 
 -3549 
 
 -2336 
 
 -1586 
 
 
 1 ' ^ 
 
 
 
 
 
 
 
 TABLE LII. 
 
 Uuit-l". 
 
 
 F . (n.-n+l)+w / 
 
 - 79. 10 
 
 
 + 142.49 
 
 + 50.72 
 
 + 23. 48 
 
 + 12. 03 
 
 
 fl . (n.-n-l)-*' 
 
 + 26. 37 
 
 + 95. 96 
 
 + 142.49 
 
 
 - 70. 45 
 
 - 24. 07 
 
 
 #,.0(71+1. -n+l)+jr' 
 
 - 609.9 
 
 - 528.9 
 
 - 231.4 
 
 - 108.8 
 
 - 52.7 
 
 - 25.7 
 
 
 Hi. (n-l.n+l)+x' 
 
 + 124. 2 
 
 + 528. 9 
 
 +1121. 6 
 
 
 - 897.4 
 
 - 365. 9 
 
 
 ffi. (w+l. wl)t / 
 
 - 530.8 
 
 
 + 551.7 
 
 + 166. 9 
 
 + 64.3 
 
 + 26.5 
 
 
 tf^n-l.-n-l)-*' 
 
 , ;i yrj 90.3 
 
 - 312.4 
 
 - 421.4 
 
 - 572. 6 
 
 - 967.8 
 
 
 
 H . l (n.-n+2)+7:' 
 
 
 +1057. 8 
 
 + 311. 5 
 
 + 111.3 
 
 + 39.4 
 
 + 11.6 
 
 
 S tl .i(n.n)-\-x / 
 
 - 166.5 
 
 -1057. 8 
 
 
 +1145. 2 
 
 + 501.5 
 
 + 276. 
 
 
 H O ,,(.-)-K' 
 
 + 166.5 
 
 + 290.1 
 
 
 + 71.9 
 
 + 62.1 
 
 + 44.9 
 
 
 H . 1 (n.-n-2)-7r / 
 
 + 162.4 
 
 + 608.5 
 
 + 881.4 
 
 +1551. 
 
 
 - 1020.6 
 
 
 H . (n.-n+l)+x' 
 
 + 406. 6 
 
 
 - 747.8 
 
 - 297.2 
 
 - 152.4 
 
 - 85.8 
 
 
 H . (n.-n 1)-*' 
 
 - 100.4 
 
 - 256.6 
 
 - 177.8 
 
 
 + 739. 1 
 
 + 219. 8 
 
 
 ^,. (n+l.-n+l)+^ / 
 
 +2402 
 
 +2483 
 
 +1362 
 
 + 740 
 
 + 406 
 
 + 222 
 
 I 
 
 Zr,. (n-l.-n+l)+,r' 
 
 - 717 
 
 -2483 
 
 -4550 
 
 
 +8048 
 
 + 3120 
 
 y 
 
 F,. (n+l.-n-l)-^ 
 
 +2046 
 + At c y 
 
 _I_1 91 A 
 
 -4336 
 
 1 T API 
 
 -1408 
 
 _i_l CMYI 
 
 - 697 
 
 11 -\ Of* 
 
 298 
 
 1 
 
 fi Q \(n _ n +2)+^ * 
 
 <6& 
 
 -f-U14 
 
 -3908 
 
 -rJ.4ol 
 -1705 
 
 -piyui 
 - 753 
 
 -plloO 
 
 - 325 
 
 - 126 
 
 
 8 9 . l (n.-n)+* f 
 
 + 893 
 
 +3908 
 
 
 -9529 
 
 -3936 
 
 - 2233 
 
 
 n^.^n.n)!^ 
 
 - 893 
 
 -1855 
 
 
 39 
 
 - 287 
 
 - 270 
 
 
 ^ . 1 (n.-n-2)-^ 
 
 - 693 
 
 -1987 
 
 -2363 
 
 - 310 
 
 
 +13643 
 
No. 3.] 
 
 MINOR PLANETS LEUSCHNER, GLANCY, LEVY. 
 
 TABLE LIU. 
 
 } Unit-l" 
 
 143 
 
 and by eq. (193), 
 
 
 Sin 
 
 UP 1 
 
 w" 
 
 10 
 
 v"*- 
 
 ^+40+3J-n' 
 
 - 25. 36 
 
 + 123.2 
 
 - 281.8 
 
 ^ 
 
 40+3J-IF 
 
 -J+20+ J-U' 
 <j>+26+M+U' 
 <!>+26+ J-U' 
 f+W+M-U' 
 
 + 50.7 
 - 816. 8 
 - 521.8 
 + 432.9 
 + 129.9 
 
 - 246. 5 
 + 3636 
 + 2851 
 - 2034 
 - 861 
 
 + 563.6 
 -8548 
 -7663 
 +5237 
 +2770 
 
 if 
 
 4-26 +n' 
 <!>+28+24+n' 
 <!>+28+2J-n' 
 0+60+4 J-n' 
 
 - 649.4 
 + 596.4 
 - 26.5 
 - 214 
 
 + 3096 
 - 2916 
 + 494 
 + 1236 
 
 -7475 
 +7216 
 -2266 
 -3395 
 
 
 (0-0 ) cos 
 
 
 
 
 
 4+ J+U' 
 
 + 191. 93 
 
 - 705. 2 
 
 +1302. 6 
 
 y 
 
 4+n' 
 
 - 383.8 
 
 + 1410 
 
 -2605 
 
 1)' 
 
 4+ A+H' 
 
 4- 4-n' 
 
 +6584 
 -5312 
 
 -40060 
 +29610 
 
 
 ril' 
 
 J>+ 2J+U' 
 
 <!>+ n' 
 4- n' 
 
 -5742 
 -6024 
 +6024 
 
 +36970 
 +38180 
 -38180 
 
 
 1" 
 
 <<,+ A+W 
 $+ J-U' 
 
 +6584 
 -1656 
 
 -40060 
 [+11860] 
 
 
 f 
 
 4+ j+n' 
 
 -3002 
 
 +18970 
 
 
 
 m' 
 
 2 
 
 By inspection it is clear that the periodic part of S is of the form 
 
 2 Up. q i)Pr]'* sin A 
 and the secular terms are of the form 
 
 o'A'^ cos {(A-t 
 
 w 
 
 . T) 17,. COS A 
 
 Expanding cos {(A s) +s}, and collecting coefficients of sin and cos e, the secular terms can 
 be written 
 
 nt{ K^cos e e) + Kj sin e} 
 where 
 
 IT, = I U p . q ijPi)'i^ cos (A E) - 4 C/j.o cos A 
 
 Introducing this notation, the perturbation can be written in the form of eq. (205). 
 
 The coefficients U p . q are given in Table LIV. K, and K 2 , which are constants, are tabu- 
 lated in Tables LV^ and LV n , respectively. For a given planet the factors and arguments are 
 known. Therefore T^ and K^ reduce each to a single numerical quantity. 
 
 Since the Bohlin-v.Zeipel method is based on the fundamental principles of Hansen, the 
 constants of integration are determined by the condition which must be satisfied when the 
 
144 
 
 MEMOIRS NATIONAL ACADEMY OF SCIENCES. 
 
 [Vol. XIV. 
 
 perturbations are developed on the basis of osculating elements, namely, that the perturbations 
 and their first derivatives shall be zero at the time t = 0. The relations to be satisfied are 
 
 u = 
 
 -0 
 dt~". 
 
 and the following equations are equivalent relations : 
 
 Logarithmic. 
 
 Unit-l" 
 
 
 Sin 
 
 M 
 
 w' 
 
 W 
 
 1) f 
 1 x 
 
 - 4-n' 
 -n' 
 
 29+ 4-n' 
 49+34 -H' 
 49+24 -n' 
 
 1.705 
 
 3. 0621 B 
 2. 8235 
 2.2831 
 3. 1591,, 
 3. 2462 
 
 3. 7258 
 3. 5528 
 2. 8483 B 
 3. 8608 
 3. 9166 B 
 
 V 
 
 j + 9 -n' 
 i+ 9+ 4-n' 
 
 i+39+24-H' 
 J:+59+44 n' 
 
 'II 
 
 1 1 ! 
 
 3. 2112 B 
 2. 5875 
 2. 2787 
 3.3155 
 3. 0779 B 
 
 3.8544 
 3. 4153 B 
 2. 6304 n 
 3. 5865 B 
 3. 3972 
 
 '<; 
 
 \e 9 24 n' 
 -}e+ 9 -n^ 
 _ j -j-39+24 n' 
 
 
 3. 1158 B 
 3. 1493 
 2. 3242 
 3. 3863 
 3. 3532 n 
 
 3. 7378 
 3. 7544 B 
 3. 0060 n 
 4. 1833 B 
 4. 1452 
 
 i 
 
 -)-29+ 4 n' 
 +29+24 -n' 
 
 +69+44 -n' 
 +69+54 -n' 
 
 2.6364 
 1.423n 
 1. 4042 n 
 2. 3306,, 
 2. 1137 
 
 3. 3704 B 
 2.706 
 2. 1720 
 3. 1922 
 3. 0138 n 
 
 3.8423 
 3. 4014 n 
 2. 6339 B 
 3. 7582 n 
 3. 6101 
 
 T\ 
 
 - -29-34 -n' 
 - -29-24 -H' 
 
 - - 4-n' 
 - +29 -n' 
 
 - +29+ 4-n' 
 
 2. 7175 
 2. 7756 B 
 
 2. 8125 
 2. 9121n 
 
 3. 4858 B 
 3.5070 
 1. 6810 
 3. 4427 B 
 3. 4958 
 
 3.9484 
 3. 9456 n 
 2. 2463 B 
 3.7846 
 3. 8338 n 
 
 '* ' 
 
 $+39+24 -n' 
 $+39+34 -H' 
 $e+59+44 -n' 
 f +79+54 -n' 
 
 
 2.6058 
 1.760 
 1. 7510 n 
 2. 9120 B 
 
 3. 5312 n 
 
 1.82 B 
 2. 8113 
 4. 0813 
 
 ''; 
 
 -f*- 9-24-H' 
 -$+ 9- 4-H' 
 
 -$+ e -n' 
 
 i ''A ' 
 
 2. 8673 
 2. 9620 B 
 2. 0569 B 
 2. 9275 B 
 2. 9702 
 
 3.8458^, 
 3. 9124 
 2. 7932 
 3. 4708 
 3. 5487 B 
 
 v 
 
 2+49+34-n' 
 2+49+44-n' 
 2+69+54-H' 
 
 
 1.640 
 1.617 
 1. 206 B 
 
 2. 731 n 
 2. 340 n 
 2. 2110 
 
 TJ 
 
 -2e-49-54-n' 
 -2i-49-44-n' 
 -2-29-34-H' 
 -2 -24 -n' 
 -2j - 4-n' 
 
 
 2. 4012 
 2. 5241 B 
 1. 5290 n 
 2. 3174 n 
 2. 3514 
 
 3. 3634 B 
 3.4544 
 2. 3210 
 3. 0558 
 3. 0737 B 
 
 
 m' 
 
 u 
 
 .=2Ur,. a i)P-n'9a\n A+nt{K,(coa e )+JsT, sin }+c,(cos t e)+c, ain E 
 
 I COS I V V 1 
 
 7 IT 
 
No. 3.] 
 
 >>! 
 
 MINOR PLANETS LEUSCHNER, GLANCY, LEVY. 
 
 TABLE LV a . 
 
 145 
 
 Logarithmic 
 
 Unit- 1". 
 
 
 
 OH 
 
 w 
 
 
 
 V 
 
 * 
 
 1 
 
 j-n' ' 
 
 j+n' 
 ^+n' 
 4+n' 
 j+n' 
 2J+n' 
 
 2. 9180 B 
 1.9821 
 2.8035 
 3. 5175 
 3. 1764 B 
 3.4580 n 
 
 3. 7732 
 2.5473, 
 3.7182,, 
 4. 3017, 
 3. 9772 
 4.2668 
 
 2.8138 
 
 
 m' 
 
 /*/* cos Arg. 
 . -A. i TABLK LV '- ;o ; 
 
 Logarithmic 
 
 Unit-l". 
 
 i j-.rl* if e.* 
 
 
 Sin 
 
 tr 
 
 u 
 
 w 
 
 . 
 
 ' i .> 1 'yls III *WJ 
 
 
 
 
 
 
 
 , .,.,() ;f.tj|,, 
 
 
 
 
 
 
 f.. viioiti -fiKt 
 
 
 V* 
 
 j n' 
 
 2.9180 
 
 3. 7732, 
 
 
 
 
 ^ 
 
 n' 
 
 3.7799 
 
 4. 5819, 
 
 
 
 
 
 4+n' 
 
 1. 9821, 
 
 2.5473 
 
 2.8138, 
 
 >i>t .(i<j(jfnr!''!> 
 
 
 'V 
 
 j+n' 
 
 3.7744, 
 3. 5175, 
 
 4.5420 
 4. 3017 
 
 
 
 
 1 v' 
 
 2J+n' 
 
 3.4580 
 
 4.2668, 
 
 i r '. *- - . i , r f 
 
 
 
 j" 
 
 j+n' 
 
 3.1764 
 
 3. 9772, 
 
 
 r,,fT .]ioll;)fl.l 
 
 
 
 n' 
 
 F! 1 ". ' :'.' ' >' \ 
 
 ITj = SvPTpy'i]* sin Arg. 
 
 1^ 
 
 i;Pj;'9 sin ^4 +7J.K, (cos s e) + K, sin e +c,(cos e e) -fc, sin e 
 
 r cos i ~ M 
 
 By eq. (205), at the date of osculation, 
 
 / = ft n = fj- 
 
 - 
 
 w 
 
 , , ' i 
 
 c,(cos e e) +c, sin s (A.) 
 
 
 t COS t 
 
 By Hansen, 1 
 
 
 
 d( u \ d( U \dS 
 d\t cos t/ = d^V cos i) d<f> U 
 
 in which v. Zeipel's notation is adopted. 
 
 dS 
 
 xi_ . ., 
 
 tne derivative, -, contains the constants 
 
 <t ^ongiateafT 
 >iiT *.i JT 
 From the various parts of S, enumerated above, *jj can be computed. Since S contains 
 
 the constants of integration 
 
 c,(cos <f> e) +0, sin ^ 
 
 .'ii -tMl . L' 'i9<i yjininqolsv^b aniwollol 'idl i<>} 
 IF! v wMi/ju-. -noil tJOTim..-> od IIAO -T;i-wn-(> -dl 
 c, sin e + c 3 cos e 
 
 The solution of eqs. (A) and (B) gives c^ and Cj. But there is a better way of deter- 
 mining the derivative of the perturbation. The exposition of this second method is 
 postponed until a particular example is considered, for the perturbations are not yet in a form 
 which leads to the development of the equations. 
 
 1 Auseinandersetzung einer iweckmassigen Methode rur Berechnung der Absoluten Stomngen der Idcinen Planeten, Erste Abhandlun, 5 5, p. S 
 110379 22 10 
 
 >,-ib 1.,'; ITO! :! ll/-.'''' 
 
146 MEMOIRS NATIONAL ACADEMY OF SCIENCES. [Voi.xnr. 
 
 COMPARISON OF TABLES. 
 
 Tables L, LI, LII check satisfactorily. 
 
 Table LIII. With one exception, the agreement is satisfactory. The bracketed coefficient 
 contains a misprint in sign in v. Zeipel's table. That it is a misprint is evident from Table LV t , 
 in which the correct sign is given to the corresponding coefficient. 
 
 The terms included in the last column are computed from the additional tables, 
 2 , XlVifl 2 and from first degree terms in Z 116, eq. (200). The latter part, namely, 
 
 e cos . 
 
 is added to both eq. (200) and eq. (203). 
 
 Table LIV. Our table is more extensive. The one bracketed quantity includes an addi- 
 tional term from Table LIII. 
 
 Tables LV I( LV U check satisfactorily. 
 
 CONSTANTS OF INTEGRATION IN ndz AND v. 
 
 The constants in . were treated in the preceding section by the familiar Hansen method. 
 cos i 
 
 It is the purpose of this section to modify the similar treatment of the constants in the per- 
 turbations ndz and v so as to incorporate them in the elements a w e , ic w ^ - Through the con- 
 stants of integration, the constant elements, which have been used from the beginning without 
 definition, are to be explained. 
 
 Since the group method of developing perturbations is built upon the fundamental prin- 
 ciples of Hansen, his conditions for the determination of the constants of integration must be 
 fulfilled. These conditions depend upon the choice of initial osculating or mean elements. 
 Osculating elements are used here. The corresponding conditions are that the perturbations 
 and their first derivatives, at the date of osculation, (< = 0), shall be zero. 
 
 Consider the relation of the constants of integration to the elements. There are two con- 
 stants in each perturbation since the differential equations are of the second order. The con- 
 stant added in the first integration is a velocity; the one added in the second integration is a 
 displacement, or, a perturbation. Now, recalling that the position and velocity of a body for 
 any time t can be transformed into the constants which are ordinarily called the elements of 
 the orbit, it is evident, by analogy, that a displacement of the body and the velocity of the 
 displacement can be transformed similarly into changes in the elements. The four constants 
 in n$z and v are related to the four elements, a, e, TT, c, defining the shape and size of the orbit 
 and the position in the orbit, and the two constants in the perturbation which is measured perpen- 
 dicular to the plane of the orbit are related to the elements fi, i, which determine the position 
 of the plane of the orbit. It is possible therefore to modify all six elements, but it is v. Zeipel's 
 preference to make the transformations only for the first four constants. 
 
 It is not necessary to compute 
 
 ndz v 
 
 k> eJificj euorutv 'fi mo 
 
 dn8z dv t = 
 
 Jflfjj- <:<)> )ti) 
 
 de de 
 
 for the following developments perform the transformation analytically, and the changes in 
 the elements can be computed from auxiliary functions. 
 
 Let a , e , x , c be osculating elements ; let a, e, K, c be the osculating elements modified 
 by the constants of integration in the manner indicated above. 
 
 For undisturbed motion, 
 
 -e sin = c + 7i < ll+^> 
 
 <sKv-7M,) = -\/nrz *9 i 
 
 r cos (v 7r ) =a (cos e e ) r sin (v JT O ) =a Vl e 2 sin E 
 
 Hansen's choice of ideal coordinates demands that the coordinates and their velocities 
 shall have the same form of expression for disturbed and undisturbed motion. The ideal polar 
 
NO. 3.] MINOR PLANETS LEUSCHNER, GLANCY, LEVY. 147 
 
 coordinates are designated by E or /and f. The relations which are analogous to the above are 
 
 !l!JMV4 i!) HO 
 
 tg(v -JT O ) 
 
 F cos/=a (cos e-e ) f sm/=a o yi -e e 2 sin I 
 
 f=vn^ r=f(l+v) 
 
 These are the equations for motion in the orbit based on constant osculating elements and 
 appropriately determined constants of integration. 
 
 If, in place of osculating elements and Hansen's ndz and v, v. Zeipel's elements and the 
 corresponding perturbations are used, the equations are the same in form. In v. Zeipel's 
 notation e and / take the place of e and /. The omission of the dash over these variables is 
 permissible, since the physically real values, with which they might be confused, do not occur 
 in the theory except for the date of osculation, where the subscript zero is added. It is to 
 be noted that, through v. Zeipel's choice of elements, the coordinates and the perturbations 
 have values which are numerically different from the Hansen quantities of the same designation. 
 
 Let the time be the date of osculation and denote the true coordinates by , v , r,. Then 
 the preceding equations for undisturbed motion become Z 121, equations (206), (207), and 
 Z 125, equation (230). 
 
 Let the disturbed eccentric anomaly and radius vector (e, r) be e, and r v respectively. 
 The relations for disturbed motion become Z 121, equation (209), and Z 122, equations (210). 
 
 The first derivatives of these expressions are given by equations (208) and (211), respec- 
 tively, and the time rate of is given by the equation following (209). 
 
 The solution of the four equations (210), (211), with the aid of all the others, determines 
 the four unknown constant elements, a, e, n, c, or, better, a a , e e , r K O , and c. 
 
 The fact that the adoption of the new elements in connection with the perturbations ndz 
 and v, as developed in the preceding sections, is equivalent to the use of osculating elements, 
 follows from the simultaneous solution of the equations for the disturbed coordinates and their 
 velocities and the corresponding equations for undisturbed motion. 
 
 The method of calculating c from the equation 
 
 c = , - e sin , - ndz 
 is given in the example, page 18. 
 
 After many laborious transformations the other three unknowns are expressed in terms of 
 familiar functions in equations (233)-(236). In the verification of these equations slight differ- 
 ences in the numerical coefficients of certain unimportant terms were found. The magnitudes 
 of these coefficients depend upon the number of the terms included in making the transforma- 
 tions. Since it makes little difference whether or not they are included and since v. Zeipel's 
 values present a more symmetrical form of a later auxiliary function, we adopted his coeffi- 
 cients. 
 
 In the functions x, y, z the arguments and factors are functions of ij, ic, ff v J, 2, where 
 
 0, = 2 ki-* sin e,)-^' 
 
 but at the beginning of the computation only T; O , r , , , J , J c , the corresponding functions of 
 osculating elements are known. 1 
 
 i There is a confusion of notation in v. Zeipel's developments. In Z 127, equation (238), Ha is denned to be the value of at the date of oscu- 
 lation when osculating elements are used for the planet, and 0\ signifies tin- argument if the elements a, t, *, etc are employed or by Z 9 
 equation (43), l 
 
 hykt-e, sin )-/ 
 
 j -l-ii.lV <)\(>l\' 
 
 and their di'Terence is computed by Z 127, equation (238). 
 In the collection of formulae by Z 133, 
 
 0- 4 c. c' 
 This is an approximation for the above equation. 
 
 Oi-ic-c' 
 
 -j- (n e sin )-# ef 
 If the secular terms are counted from the date of osculation, the factor (9 ) ought to be replaced by (0 ft). 
 
148 MEMOIRS NATIONAL ACADEMY OF SCIENCES. [Voi.xiv. 
 
 By equations Z (43), (235), (236) and the equations preceding (233), the factor ij and the 
 arguments J, e w t are given in equations (238) in terms of osculating values and functions of 
 perturbations, inclusive of first order. 
 
 To these should be added 
 
 1 i 
 2 = 2 -j-s+ 
 
 and 4ljo 
 
 -i, cos s sn e n -z cos 
 
 
 where ,1- j 
 
 
 The equations (233), (235), (236), and (238) permit the construction of two tables which 
 determine w, n or a, and e and TT. From here on the developments differ in form from v. Zeipel's 
 although they are the same in principle. If v. Zeipel's equations (237) and (239) are used, the 
 term (x," ijy/') should read 
 
 (* 2 " +x s " +x t ")-i)(y 3 " +y s " +y t ") 
 
 in agreement with Z 91, line 14. 
 
 Suppose that w w has been computed by equation (233) and the argument F has been 
 
 introduced. The arguments and factors are unknown. 
 
 .1' ')()*:) sjmwnliot 1; ' ''.'* nv/ij} 8* i io -jKM 3;m' f>rlf bn ,-.M-jnt 
 
 By Taylor's theorem ~ W =/()J ' FV &u *' ^ 
 
 df 8f df 8f df 
 
 w-*>*=f(>), r, e , J , S )+-^j, +-^jr + ^j0 +^jj +^jj + r .,,^ wj 
 
 Inclusive of second order in m', the differentiation is for first order terms. 
 
 Substituting the values of Jij, AF, J0 , JJ , JJ from equations (238) and the additional 
 equations above, 
 
 _,, ra A iMa./*/. 5 / 5 /N ! /'*/' V VV 
 
 .-/^ / , , *., ^+\f^3 5Jo ^ ;4^ 2+ VH, wiT'.WwSS! 
 
 ai i ^3/ 2 / 1 - 
 
 ,, 
 
 "'Jo COS + 1-^9 COS ,l/ Sin -3 COS 
 
 The order of calculation is: computation of equation (233), in which the arguments and 
 the factors are given the subscript zero, differentiation of first order terms, computation of the 
 second order terms in the above equation, and the additon of these second order terms to the 
 first calculation. 
 
 With some foresight the computation can be simplified. The arguments should be arranged 
 in groups like the following: 
 
 
 
 TU f u i / 
 
 ihen, for whole groups of arguments, 
 
 df _8j_ _d _ 
 ~~- 
 
 Also for some particular argument in a group, the condition 
 
 may be satisfied. 
 
Ko.8.] 
 
 MINOR PLANETS LEUSCHNER, CLANCY, LEVY. 
 
 149 
 
 Finally, by inspection of the arguments, considerable computation can be avoided if 
 
 ._ A/L a/ 
 
 The function w w> is tabulated in Table LVL Since it is unavoidably a function of w 
 itself, the determination of to for a given case must be made by successive trials, the first 
 approximation being n>=w 
 
 Logarithmic. 
 
 TABUS LVI. 
 
 w tc 
 
 Unit -1 radian. 
 
 
 Cos 
 
 ~ 
 
 ^ 
 
 - 
 
 K* 
 
 w 
 
 .. 
 
 
 
 
 4.360 
 
 [5. 1966,] 
 
 [5. 7767] 
 
 
 
 
 r 
 
 
 
 4.766 
 
 6.6599 
 
 7. 3732, 
 
 7.7492 
 
 
 zr 
 
 
 
 4.446 
 
 7.1194 
 
 7. 7572, 
 
 8.0553 
 
 
 3r 
 
 
 
 4.412 
 
 6.8442 
 
 7.5458, 
 
 7.9060 
 
 
 
 
 
 4.484 
 
 6.5883 
 
 7.3450, 
 
 7.7602 
 
 
 5/ 1 
 
 
 
 
 6.3437 
 
 7. 1490, 
 
 7. 6136 
 
 
 7r 
 
 
 
 
 5.875 
 
 6.7632, 
 
 7.3134 
 
 I'D 
 
 -5r+20 +2J 
 -4r+20 +2J 
 
 
 
 4.161, 
 
 6.5090 
 6.169 
 
 6.6325, 
 7.0658 
 
 7. 4746, 
 7.8698, 
 
 
 -3r+20 +2J 
 -2r+20 +2J 
 
 
 
 3.19 
 3.52 
 
 6. 8821, 
 7.0986, 
 
 7.6078 
 7. 6970 
 
 7.9975, 
 7.9394, 
 
 
 - r+20 +2j 
 
 
 
 5.1420 
 
 6.359 
 
 7.0722, 
 
 7.4480 
 
 
 20 +2J 
 
 
 4.379 
 
 7.6355, 
 
 8.2144 
 
 8.4125, 
 
 
 
 r+2S +2J 
 
 
 
 4.856, 
 
 8.0894, 
 
 8.9548 
 
 9.5668, 
 
 
 2r+20 +2J 
 
 
 
 4.92, 
 
 7. 8150, 
 
 8.6561 
 
 9.2006, 
 
 
 3r+20 +2J 
 
 
 
 5. 5174, 
 
 7.6056, 
 
 8.4650 
 
 9. 0111, 
 
 
 4F+20 +2J 
 
 
 
 5.4248, 
 
 7.4128, 
 
 [8. 2958] 
 
 [8. 8561,] 
 
 
 5r+20 +2J 
 
 
 
 
 7.2254, 
 
 8.1426 
 
 8.7346, 
 
 
 7r+20 +2J 
 
 
 
 
 [6. 8746,] 
 
 [7.8484] 
 
 8. 4936, 
 
 jf 
 
 -5r+20 + J 
 
 
 
 
 6.8776, 
 
 7.5604 
 
 7.8425, 
 
 
 -4r+20 + J 
 
 
 
 4.582 
 
 6.8815, 
 
 7.4536 
 
 7.5238, 
 
 
 -3r+20 + J 
 
 
 
 4.674 
 
 6.6271, 
 
 6. 7816 
 
 7. 3174 
 
 
 -2r+20 + J 
 
 
 
 4.99 
 
 6.7985 
 
 7.4732, 
 
 7. 7966 
 
 
 - r+20 + J 
 
 
 
 5.4623, 
 
 
 
 
 
 20 + J 
 
 
 4.605, 
 
 [7. 1987] 
 
 7.8314, 
 
 8.1061 
 
 
 
 r+20 + J 
 
 
 
 5.0056 
 
 8.2964 
 
 9.1086, 
 
 9.6833 
 
 
 2r+20 + J 
 
 
 
 4.38 
 
 8.0434 
 
 8.8316, 
 
 9.3296 
 
 
 sr+20 + J 
 
 
 
 5.6251 
 
 7.8458 
 
 8.6564, 
 
 9.1558 
 
 
 
 
 
 5.5812 
 
 7.6603 
 
 8.5030, 
 
 9.0248 
 
 
 5F+20 + J 
 
 
 
 
 7. 4778 
 
 8.3544, 
 
 8.9050 
 
 
 7r+20 + j. 
 
 
 
 
 7.1130 
 
 8.0545, 
 
 a6668 
 
 "to* 
 
 
 4.664 
 
 4.n 
 
 5.83 
 
 
 
 
 
 r 
 
 
 
 
 7.8102 
 
 a 6250, 
 
 
 
 2r 
 
 
 
 
 7.7520, 
 
 ai242 
 
 
 
 sr 
 
 
 
 
 7.6172, 
 
 6.6043, 
 
 
 
 4r 
 
 
 
 
 7.7135, 
 
 a2308 
 
 
 fc 1 
 
 4r+4ff tt +4J, 
 
 
 
 
 7.1862 
 
 7.9072, 
 
 
 
 _3/^_j-40 -j-4j 
 
 
 
 
 7.1804 
 
 7. 8679, 
 
 
 
 2f+40 +4J 
 
 
 
 
 6.817 
 
 7.456, 
 
 
 
 F+40 +4Jo 
 
 
 
 
 8.4680, 
 
 8.8822 
 
 
 
 40 +4J 
 
 4.666 
 
 [5. 807,] 
 
 [8. 0913] 
 
 a 8270, 
 
 9.2073 
 
 
 
 /"+40 +4J 
 
 
 
 
 a 7850 
 
 9.8236, 
 
 
 
 2r"+40 +4J 
 
 
 
 
 [a 5144] 
 
 9. 4910, 
 
 
 
 3f+40 +4J 
 
 
 
 
 a 3274 
 
 9.3006, 
 
 
 
 4/"'_j-40 -^4 < 
 
 
 
 
 8. 1627 
 
 9. 1494, 
 
 
 
 5r+40 +4J 
 
 
 
 
 8.0050 
 
 9.0105, 
 
 
 wf 
 
 _4f-(-40 4-3j o 
 
 
 
 
 7.354, 
 
 a 1083 
 
 
 
 -3^+40o+3jo 
 
 
 
 
 7.5708, 
 
 8.2084 
 
 
 
 -~r+40+3Jo 
 
 
 
 
 8.8838 
 
 9.0548, 
 
 
 
 40 +3 J 
 
 4. 516, 
 
 [6.2084] 
 
 8.5565, 
 
 9.2180 
 
 9. 5174, 
 
 
 
 r t +40 +3J 
 
 
 
 
 9.2783, 
 
 0.2833 
 
 
 
 2f +40 +3J 
 
 
 
 
 9.0241, 
 
 9.9635 
 
 
 
 3f +40 +3J 
 
 
 
 
 a 8480, 
 
 9.7850 
 
 
 
 4r+40 +3J 
 
 
 
 
 a 6916, 
 
 9.6434 
 
 
 $r +40 +3 J 
 
 
 
 
 8.5401, 
 
 9.5128 
 
 
 
 
 
 
 
 
 
 
 * 
 
 m- 
 
 m",m' 
 
 m /t , m' 
 
 m' 
 
 m' 
 
150 
 
 MEMOIRS NATIONAL ACADEMY OF SCIENCES. 
 
 [Vol. XIV. 
 
 TABLE LVI Continued. 
 
 Logarithmic. w w a 
 
 Unit- 1 radian. 
 
 
 Cos 
 
 *, 
 
 -* 
 
 M 
 
 * 
 
 
 fl 
 
 w' 
 
 -4r+ 4 
 
 
 
 
 7. 7640 
 
 7. 8364,, 
 
 m 
 
 
 ~3/ 1 + 4 
 
 
 
 
 7. 4203 
 
 8. 3915 
 
 
 
 -2r+ 4 
 
 
 
 
 7. 8104 n 
 
 8. 6268 
 
 
 
 - r+ 4 
 
 
 
 
 8. 0479 n 
 
 8. 8018 
 
 
 
 4 
 
 4. 518 B 
 
 [5. 886 n ] 
 
 [5. 70 B ] 
 
 
 
 
 
 r+ 4 
 
 
 
 
 7. 1339 
 
 7.8500 
 
 
 
 2F+ 4 
 
 
 
 
 7. 8421 
 
 8. 4293,, 
 
 
 
 q r* _i_ A 
 
 O-* i~ **o 
 
 
 
 
 7. 9669 
 
 8. 6796 n 
 
 
 
 4r+ J 
 
 
 
 
 7. 9760 
 
 8. 7576 
 
 
 I' 2 
 
 -4r+40 +24 ,:'..: 
 
 
 
 
 6.9002 
 
 7. 6938 B 
 
 
 
 -3r +400+24 
 
 
 
 
 7. 1638 
 
 7. 8502 n 
 
 
 
 -2r+40 +24 
 
 
 
 
 
 
 
 
 - r+40 +24 
 
 
 
 
 8. I860,, 
 
 8. 4016 
 
 
 
 40o+24 
 
 3.76 
 
 6. 0608 n 
 
 8. 4157 
 
 8. 9760 n 
 
 9. 1661 
 
 
 
 + r +40 +24 
 
 
 
 
 9. 1714 
 
 0. 1382 n 
 
 
 
 +2r+40 +24 
 
 
 
 
 8. 9358 
 
 9. 8333 B 
 
 
 
 +3r+40 +2J 
 
 
 
 
 8. 7718 
 
 9. 6681 B 
 
 
 
 +4r+40 +2J 
 
 
 
 
 8. 6236 
 
 9. 5372 n 
 
 
 l'* 
 
 
 3.76 
 
 5. 7516 
 
 4.7 
 
 
 
 
 
 r 
 
 
 
 
 7. 8677 
 
 8. 6727 B 
 
 
 
 2P 
 
 
 
 
 7. 8610 B 
 
 8. 2228 
 
 
 
 sr 
 
 
 
 
 8. 1026 B 
 
 8. 7296 
 
 
 ft 1 1 1 .9 
 
 4r 
 
 
 
 
 8. 1538 B 
 
 8. 8728 
 
 
 f 
 
 r 
 
 *Httt .0 
 
 
 
 7. 9418 B 
 
 8. 7337 
 
 
 
 zr 
 
 
 
 
 7. 9312 
 
 8.7154 
 
 
 
 sr 
 
 
 
 
 7. 7920,, 
 
 8. 6154 
 
 
 
 4r 
 
 
 
 
 7. 639 n 
 
 8.5001 
 
 ^ 
 
 ? 
 
 -4r+40 +34-^o 
 
 Kit > 
 
 
 
 7.446 
 
 8. 1156 B 
 
 
 
 -3r+40 +3J -|o 
 
 W J 
 
 
 
 7. 1858 
 
 7. 8677 B 
 
 
 
 I r+40o+34-^o 
 
 
 
 
 7. 6176 B 
 
 7. 9693 
 
 
 
 40 +3J 0^*0 
 
 : i<HX) 1 
 
 4.804 n 
 
 7.168 
 
 7. 9368,, 
 
 8. 3724 
 
 
 
 r+40o+34--^o 
 
 UK h 
 
 
 
 7. 7887 
 
 8. 8492,, 
 
 
 
 2r +400+34 -2 
 
 ir.\ii> o 
 
 
 
 7.448 
 
 8. 4531,, 
 
 
 
 3/^+400+3 J ^o 
 
 <.'IH<', , 
 
 
 
 7. 1976 
 
 8. 2026 n 
 
 
 
 4r+40 +34-^ (> 
 
 
 
 
 6.978 
 
 7. 9963 n 
 
 
 V 
 
 20 +24 
 
 5. 4181^ 
 
 6.292 
 
 7. 4754 n 
 
 8. 6636 
 
 
 
 
 - v , 60 +64 
 
 5. 418 B 
 
 6.292 
 
 8. 6328 n 
 
 9. 4351 
 
 
 
 i)oV 
 
 20 + 4 
 
 5.885 
 
 6. 719 B 
 
 8.5059 
 
 9. 2804 B 
 
 
 
 
 20 +34 
 
 4.974 
 
 5. 896 B 
 
 8. 0326 B 
 
 8. 1975 
 
 
 
 
 60o+54 
 
 5.935 
 
 6. 780 B 
 
 9. 2774 
 
 0. 0330 n 
 
 
 
 % ,/s 
 
 20 
 
 5.744, 
 
 6.535 
 
 [8. 3811 n ] 
 
 9. 1030 
 
 
 
 
 
 5.44 B 
 
 6.327 
 
 8. 0917 
 
 8.6300 
 
 
 
 
 60o+44 
 
 5. 919 n 
 
 6.744 
 
 9. 4432 n 
 
 0.1464 
 
 
 
 i?" 
 
 20 + 4 
 
 5.301 
 
 6. 149 B 
 
 8. 2302 
 
 9. 0152 B 
 
 
 
 
 60o+34 
 
 5.301 
 
 6. 149 n 
 
 9.1294 
 
 9. 7729 n 
 
 
 
 /** 
 
 20o+24 
 
 
 
 8.5904 
 
 9. 3492 n 
 
 9.8022 
 
 
 
 20 + 4-^o 
 
 4. 502 B 
 
 5.41 
 
 8. 1011 n 
 
 8. 8726 
 
 
 
 
 
 4. 502 B 
 
 5.41 
 
 8. 0554 n 
 
 8. 9263 
 
 
 
 JV 
 
 20 + 4 
 
 
 
 8. 5592 B 
 
 9. 3245 
 
 
 
 
 200+24-^0 
 
 4.057 
 
 5. 021 n 
 
 6.887 
 
 8. 1804 B 
 
 
 
 
 60o+44-^o 
 
 4.057 
 
 5.021 n 
 
 8. 2718 
 
 9.1021. 
 
 
 
 
 
 m" 
 
 m n 
 
 -^ 
 
 m -x 
 
 m' 
 
 m' 
 
 ww =2Cw*i)P'Tflj 1 t cos Agr., where C represents the respective coefficient. 
 
No. 3.] 
 
 MINOR PLANETS LEUSCHNER, GLANCY, LEVY. 
 
 151 
 
 Turning now to the determination of e and K, let equations (235), (236) be written in the 
 form (244), where 
 
 1 2 , 1 1 
 
 "~+'V- 
 
 Multiplying the first of these by sin ^, the second by cos </> and adding, 
 
 S sin ^ + C cos <f> = g ( 
 
 sn 
 
 coa<f>+z sin 
 
 cos <f>+z sin <l>)+-r z(y sin $ z cos ^)+ . . . 
 
 4C0 
 
 Here, again, the arguments and factors are functions of the elements a, e, JT, e, and the expansion 
 in a Taylor's series is necessary. 
 
 Let 
 
 S sin <{> + (7 cos <f>=f(i), F u U J, 2") 
 
 Then the form of Taylor's series is the same as the expression for w w , (p. 148), with the 
 following modification. Within first order quantities, 
 
 ; .; 
 
 -fi 
 
 Hence, 
 
 , F lt 6 lt A, S) = n(y cos ^ + 2 sm 
 
 1. 
 = ^ (y sin s 2 cos e) 
 
 " 
 
 
 
 
 sn 
 
 
 'o w - 
 (1 TJ n COS 
 
 l- cos 
 
 The order of computation is : calculation of 
 A-SC{.. KU 
 
 1, . ,. 
 
 -s(y cos ^+2sm^) 
 
 
 
 by inspection of the table for W, in which the arguments are to be given the subscript zero, 
 differentiation of the first order terms, calculation of the necessary products of functions of y, z, 
 and the partial derivatives, and the addition of these products to the first calculation. The 
 
 required function is given in Table LVIL 
 
 e v-t- \ ---y 
 
 
 It 07 
 
 f.8f .S ,87 S 
 
 
 n t' 
 
 
 
 0" 
 
 
 
 
 |).':T> ? 
 
 
 
 
 
 >: 
 
 
 
 
 
 
 
 
152 
 
 MEMOIRS NATIONAL ACADEMY OF SCIENCES. 
 
 [Vol. XIV. 
 
 tufj .'ii ft-d l; fW 
 Logarithmic 
 
 TABLE LVII. 
 Ssin i/r+C cos <]> 
 
 j 7; on 
 
 Unit-1". 
 
 
 Cos 
 
 ,- 
 
 UJ-J 
 
 UM 
 
 to" 
 
 w 
 
 * 
 
 
 ^- 5 r+20 +24, 
 
 
 
 8.81 
 
 1. 082, 
 
 1. 5710 
 
 1. 612, 
 
 
 4.T+200+24] 
 
 
 
 9.009 
 
 1. 2314, 
 
 1.5492 
 
 0. 989 n 
 
 
 ^ 3/ 1 +20 +24, 
 
 
 
 9.318 
 
 0.931 
 
 1.604, 
 
 1.916 
 
 
 4> 2.T+200+24, 
 
 <p- r +200+24, 
 
 
 
 9.207 
 9.711 
 
 [1. 6478] 
 1.950 
 
 2. 1070 n 
 2. 3426, 
 
 2. 2333 
 2. 3713 
 
 
 ^ +20 +24, 
 
 
 9.196 
 
 2. 1712, 
 
 2. 5678 
 
 2. 565 n 
 
 
 
 4>+ r+200+24, 
 
 
 
 9. 230 n 
 
 2. 3541, 
 
 3. 1493 
 
 3. 7107 n 
 
 
 0+2/"+20 +24) 
 
 
 
 9. 220, 
 
 1. 9114 n 
 
 2. 6867 
 
 3. 1657, 
 
 
 tl>-\-3r-\-20 -\-2d 
 
 
 
 9. 724, 
 
 1. 5372, 
 
 2. 3831 
 
 2. 8623, 
 
 
 ^+4r+20 +24, 
 
 
 
 9. 494, 
 
 1. 2544, 
 
 2. 1315 
 
 2. 6333, 
 
 
 ^+5r+20 +24, 
 
 
 
 9.100 n 
 
 1.018, 
 
 1.9034 
 
 2. 4348, 
 
 ii 
 
 ^_5r +40 +44, 
 
 
 ro wioiJ->ni 
 
 ft. 771, 
 
 1.042, 
 
 1.868 
 
 2. 357 B 
 
 
 ^-3F+40 +44^ 
 
 
 
 0.06V 
 0. 3185, 
 
 1. 723, 
 2. 1626, 
 
 2. 3515 
 2. 6961 
 
 2. 6814, 
 2. 9214, 
 
 dJ iliur 
 
 t- r+40+44^ 
 
 <!> +40 +4J 
 r + r +400+44, 
 0+2r+40 +44, 
 
 9.199 
 
 9.04, 
 
 0. 497, 
 1. 0286, 
 [2. 6172] 
 0. 7226 
 0.669 
 
 [2. 7787,] 
 3. 2379, 
 [3. 2511 n ] 
 3. 1702 
 2. 7877 
 
 [3. 0649] 
 3. 1223 
 [3. 4930] 
 4. 1580 n 
 3. 7083, 
 
 3. 0993, 
 3. 9385, 
 
 4. 9365 
 4.3605 
 
 
 0+3/ 1 +40 +44, 
 
 
 
 0.9435 
 
 2. 5117 
 
 3. 4261, 
 
 4.0450 
 
 
 ^+4.T+40o+44> 
 
 
 
 0. 5122 
 
 2. 2732 
 
 3. 2042, 
 
 
 fe 
 
 V>-5r 
 
 
 U'i \0 ', - ^. 
 
 9. 814 B 
 
 1.925 
 
 2. 634, 
 
 2.984 
 
 
 <j> 4r 
 
 
 - 
 
 0. 0434, 
 
 2. 0527 
 
 2. 6896, 
 
 2.9432 
 
 
 ip3r 
 
 
 
 0. 3541, 
 
 2.145 
 
 2. 675, 
 
 2.744 
 
 
 *ijizr 
 
 
 9.140 
 
 0. 362, 
 
 2. 1351 
 
 2. 3850, 
 
 2. 4864, 
 
 
 if, r 
 
 
 
 0.4164, 
 
 2.3504, 
 
 3. 0929 
 
 3. 5397, 
 
 
 <l> 
 
 
 9. 274 n 
 
 0. 1436, 
 
 
 
 
 
 y+ f 
 
 
 
 0. 3102, 
 
 2.497 
 
 3. 1875, 
 
 3. 5978 
 
 
 <l>-\-2r 
 
 
 9. 137, 
 
 9.918 
 
 1. 9006 n 
 
 1. 0453 
 
 2. 8834 
 
 
 <P+zr 
 
 
 
 9.465 
 
 0. 812, 
 
 2. 5218, 
 
 3.3564 
 
 
 ^+4r 
 
 
 
 9.20 
 
 1. 406 n 
 
 1.729 
 
 
 J)' 
 
 V--5r+40 +34) 
 
 
 
 9.476 
 
 1.327 
 
 1. 889, 
 
 2. 2299 
 
 
 y 4/ 1 +40 +34> 
 
 
 
 9.781 
 
 1.447 
 
 2. 1506, 
 
 2.5419 
 
 
 ^ 3/'+40 +34> 
 
 
 
 9.811 
 
 2. 1070 
 
 2. 6309, 
 
 2. 8608 
 
 
 y~ 2/*+40 +3J 
 
 
 
 0. 3489 
 
 2. 5095 
 
 2. 9557, 
 
 3. 0952 
 
 
 $ -f +40 +3 J 
 
 
 
 0. 9511 
 
 3. 3599 
 
 2. 7758 
 
 3. 9726 
 
 
 ^ +40 +34( 
 
 8.76, 
 
 [0. 158] 
 
 [2. 7932 n ] 
 
 3. 3085 
 
 3. 4526, 
 
 
 
 ^+ F +40 +34> 
 
 
 
 9.961, 
 
 3. 3609, 
 
 4. 3114 
 
 5. 0691 n 
 
 
 ^+2f+40 +34i 
 
 
 
 0.491, 
 
 2. 9943, 
 
 3. 8728 
 
 4. 4922, 
 
 
 y+3/^+40 +34( 
 
 
 
 1. 0464, 
 
 2. 7293, 
 
 3. 6067 
 
 4. 1945, 
 
 
 ^+4r+40 +3J 
 
 
 
 0. 678 B 
 
 2. 4992, 
 
 3. 3946 
 
 
 ...mfc 
 
 <!> 4r+4i 
 
 J 'HB ftfl-tj 
 
 inyift ->iJi 
 
 9.848 
 0.0792 
 
 2. 0766, 
 2. 1609, 
 
 2.712 
 2. 6968 
 
 2. 9697, 
 2. 7976, 
 
 ,t ,v to *t 
 
 ^l2r+4 
 
 Ifl'-'-'fVJIT 91 
 
 9.013, 
 
 0.3941 
 0.248 
 
 2. 157, 
 2. 0455 
 
 2.491 
 2. 7898 n 
 
 1.51 
 3. 2380 
 
 'til 1 .f)'> 
 
 ^ ^"+4i 
 
 r r 'I'tXi'ltj 
 
 
 9.901 
 
 2.584 
 
 3. 2539, 
 
 3. 6434 
 
 
 ^ +4i 
 
 
 9. 885 B 
 
 0. 8518 
 
 
 
 
 
 v^+ ^"+4t 
 
 
 
 0.1664 
 
 1.836 
 
 2.448 
 
 3. 3029 B 
 
 
 ^+2r+4, 
 
 
 9.009 
 
 9.76 B 
 
 2. 1633 
 
 2. 6170, 
 
 2. 2433 
 
 
 ^+3r+4, 
 
 
 
 9.38, 
 
 2.1064 
 
 2. 7194, 
 
 2. 9212 
 
 
 ^+4r+4, 
 
 
 
 
 1. 9892 
 
 2. 6870, 
 
 
 V 
 
 ^-5r+60 +64, 
 
 
 
 
 2.3144 
 
 2. 9730, 
 
 
 
 ^ 4r+60 +64i 
 
 
 
 
 2. 9538 
 
 3. 3785, 
 
 
 
 <j> 3/"+60 +64i 
 
 
 
 
 3. 3102 
 
 3. 5843, 
 
 
 
 V^ 2r+60 +64i 
 
 
 
 
 [3. 4970] 
 
 [3. 8423 n ] 
 
 
 
 Y *^+60o+64, 
 
 
 
 
 3. 9455 
 
 3. 7269, 
 
 
 
 <!> +60 +6J 
 
 9.95, 
 
 1. 1109, 
 
 3. 1673 B 
 
 [3. 9296] 
 
 [4. 3377 n ] 
 
 
 
 if>-\- .T+600+64) 
 
 
 
 
 3. 9144, 
 
 5. 0372 
 
 
 
 ^+2/^+600+640 
 
 
 
 
 3. 5594, 
 
 4. 5942 
 
 
 
 ^+ 3 r+60 +64, 
 
 
 
 
 3. 3121, 
 
 4. 3236 
 
 
No. 3.] 
 
 Logarithmic 
 
 MINOR PLANETS LEUSCHNER, GLANCY, LEVY. 
 
 TABUS LVII Continued. 
 S sin <{>+C coo<!> 
 
 153 
 
 Unit-l" 
 
 
 COS K- 
 
 ^ 
 
 u-t 
 
 V* 
 
 W 1C* 
 
 V 
 
 ^-5r+20 +2J. 
 
 
 
 
 2.1657 
 
 2. 7221 B 
 
 
 
 A 4/ I +20 +2 J 
 
 
 
 
 2.1255 
 
 2.8004, 
 
 
 
 ^ 3r+20 +2J 
 
 
 
 
 2.234 
 
 3. 1304 n 
 
 
 
 it 2.F+200+2 J 
 
 
 
 
 2.576 
 
 3.3804, 
 
 
 
 ^- r+20 +2j 
 
 
 
 
 3.1995 
 
 3.8325, 
 
 
 
 V> +20 +2J 
 
 0.344, 
 
 1.017 
 
 2. 689, 
 
 [3. 4822] 
 
 3.9938, 
 
 
 
 ,5+ r+20 +2J, 
 
 
 
 
 2.2480 
 
 3.2839, 
 
 
 
 ^+2r+20 +2J 
 
 
 9.45 
 
 
 3.1612 
 
 3.8424, 
 
 
 V 
 
 <i-5r-20 -2J. 
 
 
 
 
 2.700, 
 
 3.5481 
 
 
 
 ^-4T-20o-2J 
 
 
 
 
 2. 817, 
 
 3.6251 
 
 
 
 ^ &r 20 2J 
 
 
 
 
 2. 9247, 
 
 3.6905 
 
 
 
 <l> 2F 20 2 J 
 
 
 9.59, 
 
 
 3.0241, 
 
 3.7470 
 
 
 
 j>- r-20 -2J 
 
 
 
 
 3.1364, 
 
 3.8346 
 
 
 
 A -200-2J. 
 
 0.117 
 
 0.95, 
 
 2.297, 
 
 [2. 7856,] 
 
 3.6614 
 
 
 
 ^+ T -20 -2 J, 
 
 
 
 
 2.8942, 
 
 3.5604 
 
 
 
 #+2r-20 -2J 
 
 
 
 
 2. 297, 
 
 3.1129 
 
 
 fci' 
 
 <f-5r+60 +5Jo 
 
 
 
 
 2.4885, 
 
 3.1691 
 
 
 
 ^ 4/ 1 +60 +5 Jo 
 
 
 
 
 2. 976, 
 
 3.5560 
 
 
 
 ^ 3/ l +60 ~t~5Jo 
 
 
 
 
 3.6541, 
 
 3.8829 
 
 
 
 ^ 2f +60 +5J 
 
 
 
 
 [3. 9514,] 
 
 [4 1632] 
 
 
 
 ^ /"+60o~t~5Jo 
 
 
 
 
 4.3903, 
 
 40037, 
 
 
 
 ^ +60 +5J 
 
 0.295 
 
 L366 
 
 3.6364 
 
 [4. 3301,] 
 
 [46662] 
 
 
 
 i5+ /"+60 +5Jo 
 
 
 
 
 4.4005 
 
 5.4966, 
 
 
 
 c5+2/ I +60 +5J 
 
 
 
 
 4.0582 
 
 5.0612, 
 
 
 
 #+3r+60 +5J 
 
 
 
 
 3.8204 
 
 4.8027, 
 
 
 fcf* 
 
 ^-5T+2o+ Jo 
 
 
 
 
 2-426, 
 
 3.0684 
 
 
 
 ^ 4/"+20 + J 
 
 
 
 
 2.399, 
 
 3.0310 
 
 
 
 ci 3/ 1 +20o+ Jo 
 
 
 
 
 2.410, 
 
 3.1305 
 
 
 
 ^ 2r*+20 + Jo 
 
 
 
 
 2.701, 
 
 3.4602 
 
 
 
 ^ /^+20o+ Jo 
 
 
 
 
 3.2842, 
 
 3.8558 
 
 
 
 ^ +20 + J0 
 
 0.444 
 
 1.188, 
 
 3.0569 
 
 [3. 7266,] 
 
 41122 
 
 
 
 <J>-\- ^"+20o+ Jo 
 
 
 
 
 2.8541 
 
 3.5823, 
 
 
 
 ^+2r+20 + J 
 
 
 
 
 3. 2191, 
 
 3.7635 
 
 
 % ,/ 
 
 ^-5r-20 - J 
 
 
 
 
 3.1551 
 
 3.9530, 
 
 
 
 #-4r-20 - J 
 
 
 
 
 3.2454 
 3.3100 
 
 3.9948, 
 40023, 
 
 
 
 A 2/ 1 20 Jo 
 
 
 9.93 
 
 
 3.3277 
 
 3.9401, 
 
 
 
 ^- r-20 - J 
 
 
 
 
 3. 1976 
 
 3.4598, 
 
 
 
 ^ -20 - J 
 
 0.490, 
 
 1.324 
 
 3.0145, 
 
 3. 7326 
 
 42787 
 
 
 
 ^+ /" 20 Jo 
 
 
 
 
 3.3632 
 
 3.9402, 
 
 
 
 <!>+ r-20 - J e 
 
 
 
 
 2.7792 
 
 3.5224, 
 
 
 loV 
 
 ^-5r+20 +3J 
 
 
 
 
 2.2738, 
 
 2.847 
 
 
 
 <p 4r"+20 +3Jo 
 
 
 
 
 2.116, 
 
 3.0290 
 
 
 
 J> 3/^+200+3 J 
 
 
 
 
 2.5858, 
 
 3. 3787 
 
 
 
 A 2/"+20 +3 J 
 
 
 
 
 2.809, 
 
 3.5429 
 
 
 
 A- r+20 +3J 
 
 
 
 
 2.650, 
 
 3. 7297 
 
 
 
 J + r+20o+3J 
 
 9.98 
 
 0.60, 
 
 2.873, 
 
 [2.685] 
 3. 5126, 
 
 3.7980 
 4.2856 
 
 
 
 ^+2r+20 +3J 
 
 
 9.46, 
 
 
 3.3438, 
 
 41208 
 
 
 q'i 
 
 s 5-5r+60o+4J 
 
 
 
 
 L9950 
 
 2.7422, 
 
 
 
 A 4r+60 +4J 
 
 
 
 
 2.6112 
 
 3.1949, 
 
 
 
 </> 3/"+60 +4 J 
 
 
 
 
 3.0556 
 
 3.5583, 
 
 
 
 y2f +60 +4 J 
 
 
 
 
 3.7934 
 
 3. 7947, 
 
 
 
 A /'+60 +4J 
 
 
 
 
 42260 
 
 44064 
 
 
 
 ^ +60 +4J 
 
 9.98, 
 
 0. 76, 
 
 3. 5017, 
 
 41098 
 
 43552, 
 
 
 
 ifi-\- r +60 +4J 
 
 
 
 
 4.2852, 
 
 5. 3521 
 
 
 
 ^+2r+60 +4J 
 
 
 
 
 3.9567, 
 
 49249 
 
 
 ,1 
 
 ^-5r+20 +2J 
 
 
 
 
 2.5018 
 
 3.0963, 
 
 
 
 </> 4/^+200+2 J 
 
 
 
 
 2.453 
 
 3.0935, 
 
 
 
 ^ 3/^+200+2 J 
 
 
 
 
 2.4799 
 
 3. 2779, 
 
 
 
 A-W +20 +2 J 
 
 
 
 
 2.9375 
 
 3. 6294, 
 
 
 
 6- r+20 +2J 
 
 
 
 
 3.2833 
 
 3.8982, 
 
 
 
 f +2e o +2J 
 
 0.025, 
 
 0.60 
 
 2.634 
 
 3. 2781 
 
 4. 0439, 
 
 
 
 </>-\- /^+2^o+2Jo 
 
 
 
 
 3.5607 
 
 4 2381, 
 
 
 
 #+ 2 r+20.+2J. 
 
 
 
 
 3. 4629 
 
 4. 1704 n 
 
 
154 
 
 Logarithmic 
 
 MEMOIRS NATIONAL ACADEMY OF SCIENCES. 
 
 TABLE L VI I Continued. 
 8 sin <j>+ C cos <}> 
 
 [Vol. XIV. 
 
 tJnit-l". 
 
 
 Cos 
 
 -. 
 
 ^ 
 
 -, 
 
 - 
 
 w 
 
 v>* 
 
 T)" 
 
 cj-5r-20 
 
 
 
 
 3.0090 B 
 
 3. 7477 
 
 
 
 ^-4r-20o 
 
 
 
 
 3. 0676 B 
 
 3. 7445 
 
 
 
 <l>sr 20 
 
 
 
 
 3. 0764 n 
 
 3. 6664 
 
 
 
 ^-2r-20 
 
 
 
 
 2. 958 n 
 
 3. 3121 
 
 
 
 ^- r-20 
 
 
 
 
 3. 1140 
 
 4. 0201 B 
 
 
 
 <l> -20 
 
 0.305 
 
 1. 127 n 
 
 2.912 
 
 3. 5491 B 
 
 3.9085 
 
 
 
 ^+ r-20 
 
 
 
 
 3. 0396 B 
 
 3. 6320 
 
 
 
 ^+2r-20 
 
 
 1 
 
 
 2. 4706 B 
 
 3. 2330 
 
 
 f 
 
 ^-5r+60 +54-.?o 
 
 
 
 
 2.006 
 
 2. 7505 n 
 
 
 
 <l>4r +60 +54 ^o 
 
 
 
 
 2.335 
 
 2. 981 B 
 
 
 
 <t> Zr +60 +54 2o 
 
 
 j 
 
 
 2.544 
 
 3. 1436 B 
 
 
 
 ^ 2.T+600+54 (> 
 
 
 
 
 2.718 
 
 3. 2445 B 
 
 
 
 v ^*+60Q+5^o *o 
 
 
 
 
 2.970 
 
 2. 9116 n 
 
 
 
 dt +600+5.^0 ^*o 
 
 8.6 B 
 
 9.7 
 
 2. 114^ 
 
 2.923 
 
 3. 4067 n 
 
 
 
 d>-}- F +60 +54 ^o 
 
 
 
 
 2. 7948 B 
 
 3.9420 
 
 
 
 ^-j_2r+60 +54 2 
 
 
 
 
 2. 3824, 
 
 3.4488 
 
 
 P 
 
 ^-5r +20 +24 
 
 
 
 
 9.6 
 
 2.387 
 
 
 
 <j> 4r+20 +24 
 
 
 
 
 1. 916 B 
 
 2.911 
 
 V ? irv 
 
 
 <!> 3r+20 +24 
 
 
 
 
 2. 5178 B 
 
 3.3047 
 
 
 
 J> 2J rl +20 +24 
 
 
 
 
 2. 938 B 
 
 3.6294 
 
 
 
 4,- r+20 +24 
 
 
 
 
 3. 3406 B 
 
 3. 9330 
 
 
 
 <{, +200+24 
 
 
 0. 5910 
 
 3. 1266 
 
 3. 8021 B 
 
 4.1894 
 
 
 
 ^+ r+20 +24 
 
 frit<> .}; 
 
 
 
 3. 4070 
 
 4. 3178 B 
 
 
 
 0+2r+20 +24 
 
 
 
 
 3.0472 
 
 3. 9308 B 
 
 
 f 
 
 0_5/--20 - 4+^o 
 
 
 
 
 0. 732 n 
 
 1.085 
 
 
 
 d>4r26 4+-^o 
 
 
 
 
 0.35 
 
 1. 895 n 
 
 
 
 <{izr20 4+^0 
 
 
 
 
 1.463 
 
 2. 5146 n 
 
 \t ,,jr 
 
 
 </r-2r-20 - 4+^o 
 
 
 
 
 2.064 
 
 3. 0255 n 
 
 
 
 ^- r-20 - 4+^0 
 
 
 
 
 2. 6816 
 
 3. 6280 n 
 
 
 
 -20 - 4+^0 
 
 9.04 
 
 0.11 B 
 
 2.636 
 
 3. 3284 B 
 
 3. 7399 
 
 
 
 ^+ r-20 - 4+J 
 
 
 
 
 3. 0572 B 
 
 3.6430 
 
 
 
 V-+2r-20 - 4+^o 
 
 
 
 
 2. 9121 n 
 
 3.5491 
 
 
 lo 8 
 
 4, +40 +44 
 
 0.775 
 
 1.66, 
 
 3. 1052 B 
 
 3. 0342 B 
 
 
 
 
 J +80^+84 
 
 0.29 B 
 0.65 
 
 1.10 
 1.54 n 
 
 3. 1888 
 3. 7520 
 
 3. 6104 n 
 4. 5812 B 
 
 
 V* 
 
 Vl' 
 
 V> +40 +54 
 
 
 
 3. 7577 
 
 4. 3244 n 
 
 
 
 
 <!> +40 +34 
 
 1.260 B 
 
 2.081 
 
 3. 1240 
 
 4. 1388 
 
 
 
 
 d> -40 -34 
 
 1.005 
 
 1.77 B 
 
 3. 5356 B 
 
 3. 3560 
 
 
 
 
 <!> +80 +7J 
 
 1. 228 B 
 
 2.093 
 
 4. 3980 n 
 
 5. 1827 
 
 
 
 lo V 
 
 ^ +400+44 
 
 
 
 4. 1155 B 
 
 4.5547 
 
 
 
 
 +400+24 
 
 1.106 
 
 1.88 B 
 
 2.831 
 
 4. 1803 B 
 
 
 
 
 <!> -40 -24 
 
 1. 146 B 
 
 1.88 
 
 3.0422 
 
 4. 0180 
 
 
 
 
 ^ +800+64 
 
 1.321 
 
 2. 152 B 
 
 4. 5658 
 
 5. 3010 n 
 
 
 
 if* 
 
 <j> +400+34 
 
 
 
 3. 8375 
 
 4. 0446 B 
 
 
 
 
 <l> -40 - 4 
 
 1 f'- ! 
 
 iiii 'i 
 
 3. 2197 
 
 3. 9650 B 
 
 
 
 
 <[> +80 +5J 
 
 
 
 4. 2553 n 
 
 4. 9349 
 
 
 
 jj, 
 
 d, +40 +34-.r 
 
 
 
 3.0024 
 
 3. 8634 n 
 
 
 
 
 df -400-34+^0 
 
 9. 98 B 
 
 0.8 
 
 2. 956 B 
 
 3. 8331 
 
 
 
 
 J, -j.g^o+74 ^"0 
 
 
 
 3. 0757 
 
 3. 9759 B 
 
 ,'{.' -. 
 
 
 
 d, +40 +44 
 
 0.46 n 
 
 1.32 
 
 3. 8514 W 
 
 4.6436 
 
 
 
 }' *>' 
 
 d> +40 +44-JT 
 
 
 
 2.442 
 
 1.846 B 
 
 
 
 
 <l> -40 -24+J 
 
 
 
 3. 2486 
 
 4. 0585 B 
 
 : 
 
 
 
 d, +80 +64-l- 
 
 
 
 3. 2818 n 
 
 4. 1441 
 
 * V'S'H ' 
 
 
 
 <!> +40 +34 
 
 0.27 
 
 1. 15 n 
 
 3.9421 
 
 4. 6972 n 
 
 
 
 S sin <p-\-C cos (/'=SC l 1 u'*7jP7/9j 2 ' cos Arg, where C^ represents the coefficient. 
 
 
No. 3.] 
 
 MINOR PLANETS LEUSCHNEE, GLANCY, LEVY. 
 
 155 
 
 COMPARISON OF TABLES. 
 
 Table LVI. Unless there are errors of calculation, all the discrepancies are due to the 
 accumulation of other discrepancies already discussed. Without going into the details of the 
 construction, it is sufficient to remark that our table is built from practically all of the available 
 auxiliary material. Our table includes many more terms than v. Zeipel's table, but it is wanting 
 in the two arguments 6F and SF in the first block of terms. These arguments contain 3s and 4e, 
 respectively, and our series were not inclusive of these higher multiples. It would be more 
 consistent to include them, since the argument 7F is included. 
 
 Table LVTI. Unless there are errors of calculation, all the discrepancies are due to the 
 accumulation of discrepancies already discussed. Our table is built from practically all the 
 available auxiliary material. Large disagreements are to be explained by v. Zeipel's use 
 of the formula following Z 131, equation (244). In this equation the following functions are 
 omitted: 
 
 cos 
 
 sn - 
 
 cos 
 
 sn 
 
 ERRATA 1 IN H. v. ZEIPEL, ANGENAHERTE JUPITERSTORUNGEN FtJR DIE HECDBA-GRUPPE. 
 
 With the exception of $ 6, Stdrungen des Radius-vector, all the developments have been checked. 
 
 Page. 
 
 Line.' 
 
 For 
 
 Read 
 
 
 
 dQ 
 
 
 1 
 
 8a 
 
 
 
 
 
 dx 
 
 &x 
 
 
 
 dQ 
 
 dQ 
 
 1 
 3ff 
 
 9a 
 
 dj 
 J 
 
 dv 
 
 
 
 /Ho 
 
 ~dT 
 
 5 
 9 
 
 5b 
 2a 
 
 (15) 
 iW 
 
 (16) 
 3U 
 
 9 
 
 Ib 
 
 W+v 3 
 
 W+r* 
 
 12 
 
 2a 
 
 w 
 
 fY 
 W 
 
 12 
 
 9a 
 
 a+ 
 
 
 
 
 n+i 
 
 P n+i 
 
 12 
 
 lOa 
 
 0* 
 
 /3 () 
 
 
 
 
 ' 
 
 12 
 
 6b 
 
 (2n+4t+l)V B (4i+4)-y < ' 
 
 (2n+4i+l)Y < l -+(4t+4)7^ 
 
 13 
 
 5a 
 
 (2n+4f+3) T< '-(4i+4)- x ^ 1 
 
 (2n+4t+l) 7< -+(4t+4)7'-* 
 
 14 
 
 2b 
 
 n'g> 
 
 ng' 
 
 
 
 
 noo 
 
 15 
 
 8bff 
 
 I 
 
 
 
 
 
 _ (j 
 
 B 
 
 16 
 
 lOa 
 
 sin 
 
 COS 
 
 16 
 
 6b 
 
 1 dQ 
 7 a *3I 
 
 1 dQ 
 
 19 
 
 Ga 
 
 dF 
 
 dF_ 
 
 
 
 da a 
 
 <fa 
 
 20 
 
 5b 
 
 TJ-? 
 
 T** 
 
 21 
 
 4a 
 
 ^.ro.n 
 
 1?j 3 - 
 
 21 
 
 4b 
 
 Metoden 
 
 Methoden 
 
 24 
 
 9b 
 
 2P ' . 
 
 2pi-. a 
 
 27 
 
 21a 
 
 P . fn 1. n+ljj, 
 
 /Vofn-?.-n+lJ_i 
 
 34 
 
 21a 
 
 P jtfi \ 7l + l)_j 
 
 
 42 
 44 
 
 5b 
 
 18b 
 
 /V 2 (n.-n) 
 .ffi.,(n+l.-n+l) 
 
 Fl'^n.-n) 
 
 45 
 46 
 
 20a 
 8a 
 
 g(n 
 
 l G (n 
 
 46 
 
 lOa 
 
 I. O (TI ! n+2) i 
 
 O ..(n ! Ti+2) t 
 
 46 
 
 lla 
 
 ,. (n 1- n)+t 
 
 O .i(nl-n)-t 
 
 1 Inclusive of those tabulated by v. Zeipel. 
 
 ' The number of the line counting from the top of the page is indicated by a, counting from the bottom of the page by b. 
 
 On page 3 and all following pages ' is denned by ' The error consists in the omission of a statement announcing a change of notation. 
 See definition of >' given on page 2. 
 
156 MEMOIRS NATIONAL ACADEMY OF SCIENCES. 
 
 Errata in H. v. Zeipel, Angenahcrte Jupittrstarungen fur die Hecuba-Gruppe Continued. 
 
 [Vol. XI V_ 
 
 Page. 
 
 Line.' 
 
 For 
 
 Read 
 
 
 
 J>Q 
 
 dfl 
 
 49 
 
 7b 
 
 
 f^ 
 
 
 
 Os 
 
 dr 
 
 50 
 
 6b 
 
 r 2 
 a 2 
 
 r ''iimytxi owj orii a 
 a 2 
 
 50 
 
 6b 
 
 3+ij 2 
 
 3+14, 2 
 
 51 
 
 lb 
 
 5 .,(n.-n+l) 
 
 
 53 
 
 lib 
 
 >; 
 
 >" 
 
 54 
 
 5a 
 
 E 
 
 -i 
 
 56 
 
 4b 
 
 
 
 61 
 
 lib 
 
 20+20 
 
 20+24 
 
 62 
 
 17a 
 
 ^+60+40 
 
 ^-)-60+4J 
 
 62 
 
 5b 
 
 +436 
 
 +439 
 
 63 
 65 
 
 9a 
 3a 
 
 f(l-ecos)TF' 2 ] 
 (106) 
 
 t(i-3oo)F / ] 
 
 (106a) 
 
 65 
 
 5a 
 
 (106) 
 
 (106a) 
 
 68 
 
 3a 
 
 W * W~ 3 tti~ J W~ l V) 
 
 tu~ 4 w"" 3 tt>~" 2 it*" 1 w t^ 
 
 69 
 
 6b 
 
 sin A 
 
 77P7j /C! j 2 'sin A 
 
 69 
 69 
 
 5b 
 4b 
 
 sin (A <l>+t) 
 sin (A-\-(f>c) 
 
 Tjpr] vj sin ^vi y-pj 
 7)P73 9?^ sin ^^4 -4-(A ^ 
 
 70 
 
 lb 
 
 W t '" 
 
 w 
 
 70 
 
 lb 
 
 cos A 
 
 TfPl)'9f* COS ^1 
 
 71 
 
 7a 
 
 ess A 
 
 -rjp-rfqft cos ^. 
 
 75 
 
 15a 
 
 4o- 
 
 4o-2 
 
 75 
 
 18a 
 
 4o-i 
 
 
 75 
 
 2b 
 
 
 4H 
 
 75 
 
 lb 
 
 AI-O 
 
 
 79 
 
 lOb 
 
 e 
 
 ^< 
 
 81 
 
 8b 
 
 1ecoB c) 
 
 (1 e cos ) 
 
 83 
 
 12a 
 
 +3744 
 
 +3344 
 
 86 
 
 4a 
 
 (128j) und (130) 
 
 (1282), (128 3 ) uod (130) 
 
 86 
 
 6a 
 
 o W^ 
 
 Ot? 
 
 T^ 1 
 
 91 
 
 9a 
 
 e COB* 
 
 e cos 
 
 91 
 
 lla 
 
 TP 2 
 
 ffj' 
 
 92 
 
 3a 
 
 1J, COS C 
 
 y, COS 
 
 92 
 
 lOb 
 
 -i(l-e cos ) (IF-Jff) (W+iB) 
 
 -J[(l- cos ) (W-IS) (W+iS)] 
 
 92 
 
 4b 
 
 ^ 
 
 ^ 
 
 93 
 
 lOa 
 
 sin A 
 
 yPij'Qj 2 * gin A 
 
 93 
 
 lOa 
 
 2' 
 
 J 
 
 94 
 
 19b 
 
 dW 
 
 dF 
 
 97 
 
 15a 
 
 (156) 
 
 (154) 
 
 99 
 
 4b 
 
 T)W sin t) 
 
 s ' 
 
 100 
 
 5a 
 
 A' 
 
 ^ 
 
 100 
 
 6a 
 
 Ju^ 
 
 _lj5 
 
 115 
 116 
 
 4b 
 
 7a 
 
 8SJ> -I 
 
 (192) 
 
 
 116 
 
 lOb 
 
 /o ro i\ 
 
 ~-j- ^Oj l^lj/ 
 
 
 119 
 
 (i) 
 
 4-? 
 
 f7p. 
 
 122 
 
 3a 
 
 
 
 123 
 
 4b 
 
 (i f 2 f) 
 
 (1 f 2 f 8 ) 
 
 125 
 
 3a 
 
 1 o COS 2 
 
 1 COS 
 
 128 
 
 7a 
 
 A 
 
 4ft 
 
 129 
 
 5a 
 
 fi 
 
 P 
 
 131 
 
 7b 
 
 
 (0 jl ) 
 
 131 
 
 6b 
 
 $+A+l) 
 
 (^_)-^4_ ) 
 
 132 
 
 8a 
 
 2.9227 B 
 
 1.9227 n 
 
 132 
 
 26a 
 
 5.3376 
 
 5.0376 
 
 134 
 
 9a 
 
 9H4Q 
 
 (4t>+/ 4 ) 
 
 135 
 
 lOa 
 
 w 
 2 
 
 T 
 
 135 
 
 lla 
 
 to 
 
 2 
 
 I 
 
 140 
 
 141 
 
 26a 
 6a 
 
 [nSz] 
 0*8998 
 
 Mel. 
 
 0/8998 
 
 i Th number of the line counting from the top of the page Is indicated by a, counting from the bottom of the page by b. 
 
'No. 3.] 
 
 MINOR PLANETS LEUSCHNER, GLANCY, LEVY. 
 
 157 
 
 ERRATA IN KARL BOHLIN. SDR LE DfiVELOPPEMENT DBS PERTURBATIONS PLANETAIRES, 1-7, 
 
 AND TABLES I-XX. 
 
 Page. 
 
 Line.- 
 
 For Read 
 
 3 
 
 5a 
 
 /( , =(1+m)a , 
 
 -. 
 
 /.'=(! +m)a 
 
 
 
 dW 
 
 
 dW 
 
 11 
 
 Ib 
 
 
 
 
 14 
 
 lla 
 
 14V 
 
 
 l-r j 
 
 20 
 
 3a 
 
 +i s cos 2* 
 
 
 -i^ COB 2 
 
 29 
 
 8b 
 
 r* 
 
 
 y'-n 
 
 29 
 
 2b 
 
 V I/' 
 
 
 e -V-i/' 
 
 30 
 
 lla 
 
 a 
 ? 
 
 
 a' 
 ? 
 
 30 
 
 lla 
 
 e' 
 
 X? 
 
 
 +p 
 
 30 
 
 12a 
 
 2n+m-l 
 
 
 2n+m+l 
 
 30 
 
 13a 
 
 /?T\* 
 
 f.-. n wi- 
 
 +'. ^jg -r g 
 
 30 
 
 13a 
 
 2n+7n 1 
 
 --!.- . l-.'v.nW 
 
 2n+m+l 
 
 30 
 
 13a 
 
 2n+wi-2 
 
 
 2n+7n+2 
 
 30 
 
 14a 
 
 2n+m-l 
 
 
 2n-i-"i-|-l . ' 
 
 30 
 
 5b 
 
 e V-i(*-O 
 
 
 e ^in( r ^) 
 
 33 
 
 9a 
 
 r*2i+ */_, i _, \ 
 10 \ * "/ 
 
 
 Xjf*U 1. ) 
 
 35 
 
 4a 
 
 (73) 
 
 
 (74) 
 
 36 
 
 la 
 
 2/V'"e'^~ 1 <- T > 
 
 "*|+|Y 
 
 2f 4 * " V IO 
 
 36 
 
 lOa 
 
 ^.,(n-l.+n+l) 
 
 
 ^..(n-l.-n+l) 
 
 38 
 
 13a 
 
 a 
 
 
 a 
 
 38 
 
 lOb 
 
 e J^lr- w>) 
 
 
 
 38 
 
 3b 
 
 2(ij')y'- 1 
 
 
 2/ . /\|_i 
 
 38 
 
 2b 
 
 2n)y'- 1 
 
 
 2(jj)y'-l 
 
 40 
 
 3a 
 
 K (n. 1 n) 
 
 
 JP / n 1 ji\ 
 
 41 
 
 13a 
 
 a 
 
 VO 
 
 a'" 
 
 45 
 
 2a 
 
 iT . (0.-n) 
 
 
 i" . (n. n) 
 
 45 
 
 9b 
 
 a 
 
 
 
 4A 
 
 7n 
 
 r(r 1)V~' 
 
 " : |."'- 
 
 , T(rl)*x f ~ l 
 
 ^O 
 
 i 
 
 1.1.2 
 
 
 h 1.1.2 
 
 46 
 
 7b 
 
 (n-)(n-t+2) 
 
 ,'V- 
 
 (n *)(n +2) n _f^ 
 
 2 
 
 2 ' r ^ 
 
 46 
 
 5b 
 
 (n-3) 
 
 
 (n ) 
 
 46 
 
 4b 
 
 v 
 
 
 n' 4 
 
 48 
 
 14a 
 
 P'jK/n 2. n) 
 
 
 P',. (n-2.-n) 
 
 48 
 
 7b 
 
 pi . \n-\-\. n 2] 
 
 
 P*,.- ITI-)-!. n 2! 
 
 48 
 
 7b 
 
 pi (n+1 n 2 
 
 
 pi (n+1 n 2) 
 
 48 
 
 6b 
 
 P^'.iin-l.-n-Zi 
 
 
 P',.,|n 1. n 2| 
 
 50 
 
 5a 
 
 R 1 .~(n-^-l.n\ 
 
 _ T 
 
 /fi -(n+1. n I)!/ 
 
 50 
 
 9b 
 
 R l \,o(nl.n+l 
 
 +1- 
 
 R l \ < ,(n\.n-\-\)+tf 
 
 50 
 
 3b 
 
 Rig-fn. n+2^+1* 
 
 
 R l (n n+2)+ / 
 
 51 
 
 Ib 
 
 P'(n+r.-n+* i 
 
 
 piln-f-r. n+^ 
 
 59 
 
 5a 
 
 fi 3 ' 1 , jn+l. n] 
 
 
 ^J3-l _Tjj 1. 71] 
 
 59 
 
 8a 
 
 ^ 3>l o-i[ n -~ n +l] 
 
 
 J-l e r n __ B ^-i] 
 
 60 
 
 6a 
 
 if 
 
 
 / 
 
 60 
 
 8a 
 
 (See footnote z ) 
 
 
 
 ftf\ 
 
 QK 
 
 ("^/t ) 
 
 /N 
 
 ~T~\7n.ftf 
 
 DV 
 
 VO 
 
 24 
 
 V^ 
 
 24 
 
 60 
 
 fib 
 
 jX""**) 
 
 
 ( +) 
 
 61 
 
 7b 
 
 P .,[n.+n+l] 
 
 
 P .i[".-n+l] 
 
 61 
 
 5b 
 
 Poit n -+ n -lJ 
 
 
 P In n 11 
 
 62 
 
 la 
 
 n P- 
 
 
 n /i 
 
 
 
 6 
 
 
 6 
 
 62 
 
 7a 
 
 P,., (n+2.-n-l) 
 
 
 P 2 .,(n+2.-n+l) 
 
 62 
 
 7a 
 
 ^2^ 
 
 
 2 
 
 62 
 63 
 
 8a 
 5a 
 
 pJ.'o(n-l'.-n+l)H 
 
 - 
 
 P.Jn+l.-n+l] 
 P . (n-l.-n+l)_* 
 
 63 
 63 
 63 
 63 
 
 lla 
 13a 
 14a 
 5b 
 
 Pj'o M-L-n-1]-! 
 PO-O n-l.-n-lj-j 
 RO-O n. n+lj-r' 
 
 P 1 . (n+2.-n-l)+, 
 P . c fn-l.-n+l]_* 
 P . n-l.-n+l]_* 
 P.o-0 n.-n IJ-r' 
 
 63 
 
 2b 
 
 
 
 RO-O i. n 1] i* 
 
 63 
 
 Ib 
 
 R . n._n !]-' 
 
 
 
 R . n. n 1]-^ 
 
 1 The number of the line countin? from the top of the pase is indicated by a. counting from the bottom of the page by b. 
 i The argument a is defined first by eq. (31), p. 20, secondly by eq. (105), p. 60. The first of these definitions is used in J 8. 
 
158 MEMOIRS NATIONAL ACADEMY OF SCIENCES. 
 
 Errata in Karl Bohlin, Sur le Developpement des Perturbations Planitaires, 1-7, and Tables I-XX Continued 
 
 Page. 
 
 Line." 
 
 For 
 
 Read 
 
 64 
 64 
 
 6a 
 12a 
 
 .Ro-o[".--l]-*' 
 
 R 2 . [n.-n+l]+ x ' 
 
 64 
 64 
 66 
 
 14a 
 15a 
 7a 
 
 (n-n) 
 #,.,[7+l. -]_ 
 
 F (n+r. -n+s 
 
 &r T "* 
 
 Ftn+r.-n+t) 
 
 66 
 
 8a 
 
 G (n+r. n+s 
 
 G (n+r. -n+s) 
 
 
 
 +3 
 
 +3 
 
 70 
 
 la 
 
 -3 P,. 2 (n+l.-n-2) 
 
 -2 Pj. 2 (n+l.-n-2) 
 
 
 
 2 
 
 -2 
 
 71 
 
 4a 
 
 F t (n. n+l)+ 
 
 F t a ( n _ w _)-i) + . 
 
 
 
 +3 
 
 +3 
 
 71 
 
 9b 
 
 -2 P lf0 (n.-n+l)-3 
 
 -2 P,. (n.-n+l)_* 
 
 
 
 2 
 
 -2 
 
 73 
 
 2a 
 
 F 2 . u (n.n+l)- x ' 
 
 .F 2 . (n.-w-l)-,r' 
 
 73 
 
 18a, ff. 
 
 See foot note. 2 
 
 f 
 
 73 
 
 4b 
 
 R . (n. n+l)+ x 
 
 R . (n.-n+l)+ x > 
 
 73 
 
 74 
 
 3b 
 8a 
 
 R . n.-n+l)+ x > 
 
 R . (n.-n+I)+ x ' 
 
 75 
 
 75 
 
 la 
 lib ' 
 
 jOn.-n+l)-,' 
 
 G . (n+l.-n)+,+j 
 
 78 
 
 Ib 
 
 To 1 '" 
 
 Ti 1 '* 1 
 
 79 
 
 Ib 
 
 fjTn+i.n 
 
 y, m-f2.n 
 
 79 
 
 *) 
 
 n=l 
 
 n 
 
 80 
 
 9b 
 
 T(+i m ' n 
 
 ^ <+j m.n 
 
 81 
 
 12a 
 
 3 
 
 
 81 
 
 13a 
 
 (120) 
 
 (120)*) 
 
 135 
 
 7a 
 
 -^D-3 1 '* 
 
 jf i. 
 
 139 
 
 3a 
 
 /y** 
 
 2ri'- 
 
 140 
 
 2a 
 
 /Y -n 
 
 2r 1 '- 
 
 154 
 
 la 
 
 (86) 
 
 (93) 
 
 161 
 
 la 
 
 -or 
 
 or 
 
 169 
 
 8b 
 
 
 
 3 
 
 
 
 I 
 
 a 
 
 170 
 
 3a 
 
 2S 2r<3< * 
 
 2T<'- 
 
 170 
 
 4b 
 
 ^ 
 
 i 2 ^ 3>B 
 
 171 ff. 
 
 
 See foot note.' 
 
 
 185 
 
 2a 
 
 3. 27886 
 
 3. 27887 
 
 185 
 
 13b 
 
 4 
 
 3 
 
 188 
 
 6b 
 
 2. 017 3 n 
 
 2. 01703,, 
 
 189 
 
 14a 
 
 3. 27886 
 
 3. 27887 
 
 197 
 
 16a 
 
 0. 146128 B 
 
 1. 146128 n 
 
 197 
 
 18b 
 
 1. 505151 
 
 1. 505150 
 
 198 
 
 15b 
 
 1. 662759 n 
 
 1. 662758 n 
 
 198 
 
 2b 
 
 0. 477121 
 
 0. 477121 n 
 
 1 The number of the line counting from the top of the page is indicated by a, counting from the bottom of the page by b. 
 1 The space between lines 18 and 19 should read }'. 
 
 ' Tables XII, XIII. XIV give the same coefficients in numbers as Tables XVI, XVII, XVIII give in logarithms, respectively. The same factor 
 should therefore occur in the former. 
 
 o 
 
 !-"